•
BJOMATHEMATICS TRAINING PROGRAM
The Locally Optimal Combination of Certain
Multivariate Test Statistics
by
Katherine Lane Monti
May 15, 1975
Institute of Statistics Mimeo Series /I
1(;)
t:J"
MONTI, KATHERINE LANE. The Locally Optimal Combination of Certain
Multivariate Test Statistics. (Under the direction of
PRANAB KUMAR SEN and DANA QUADE.)
Most overall tests of hypotheses combining s
~
2 independent
statistics are based on the p-values of the individual tests.
Van Zwet
and Oosterhoff considered procedures for combining tests of univariate
hypotheses which are based on the values of the (asymptotically exponentially distributed) statistics themselves.
This dissertation deals with
combinations of certain multivariate test statistics with distributions
which are not in the
expone~tial
family • .
Many of the classical GLMM analysis of variance statistics have
F distributions and can be transformed tIJ beta-distributed random
variables.
It is shown that if the statistics to be combined have beta
distributions which are noncentral under the alternative hypothesis,
then a type D test, which provides a locally most powerful combination
procedure for testing the null hypothesis does exist.
The form of the
test is found for the balanced case where the design matrices are proportiona1.
For the unbalanced case, a test which has the best average
local power is found; this test is termed a type F test.
It is shown that, for local alternatives, if there are a large
number of small-sample statistics to be combined, the type D comb ination procedure is more efficient than Fisher's method according to
Bahadur asymptotic relative efficiency.
It is also shown that under
the same assumptions, the type F combination procedure is more efficient than Fisher's according to an average measure of Bahadur
asymptotic relative efficiency.
The exact distribu.tion of the overall type D and type F statistics is given if s = 2.
As s increases, the exact distribution of the
statistics becomes more difficult to calculate.
The Central Limit
Theorem does apply, however, and the accuracy of applying the Central
Limit Theorem to find approximate critical values for the overall test
statistic is discussed.
The method of applying the type D and type F tests is
illustrated in a numerical example based on some pharmaceutical data.
THE LOCALLY OPTIMAL COMBINATION OF CERTAIN
MULTIVARIATE TEST STATISTICS
by
Katherine Lane Monti
A Dissertation submitted to the faculty of
the University of North Carolina at Chapel
Hill in partial fulfillment of the requirements for the degree of Doctor of Philosophy
in the Department of Biostatistics.
Chapel Hill
1975
Approved by
I
f
ACKNOWLEDGMENTS
The author appreciates the assistance of the members of her
advisory committee, Drs. J. C. Cornoni, J. E. Grizzle, R. R. Kuebler,
D. Quade, and P. K. Sen.
Special thanks go to Dr. P. K. Sen for his
guidance of this research and his reassuring support throughout the
duration of this undertaking, and to Dr. D. Quade, who adopted the
responsibilities of adviser when Dr. Sen was on sabbatical, for his
patient assistance.
Thanks are also due to Drs. T•. Donne11y and
J. E. Grizzle, who were responsible for the author's initial interest
in biostatistics.
The financial assistance of the Department of Biostatistics
during the course of the author's graduate studies is gratefully
acknowledged.
This research was supported by National Institues of
Health, Division of General Medical Sciences Grant Nos. 5TDl GM00038-20
and 5T01
GM0003~-21.
The author wishes to thank her husband, Danny, for his continued
support and encouragement throughout the course of the author's studies
and especially for his understanding while the author was preparing for
and taking the Basic Examination.
For her typing of the manuscript, Mrs. Gay Hinnant is. warmly
thanked.
ii
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS.
....
ii
LIST OF TABLES • •
v
LIST OF FIGURES.
vi
Chapter
I. INTRODUCTION.
1
1.1. Statement of the Problem and Outline
of the Research. • • • • • • • • •
1.2. Review of the Literature
••••
1.3. Introduction to Type D and Type F
Critical Regions • • •
1
4
9
1.3.1. Type D Critical Regions •
1.3.2. Type F Critical Regions ••
9
14
II. TYPE D CRITICAL REGIONS FOR COMBINING BETADISTRIBUTED STATISTICS ••
17
17
17
2.1. Introduction • • • •
2.2. Permissible Tests and Critical Regions
2.3. The Existence of a Type D Critical Region
for Combining s Independent Statistics
with Beta Distributions • . • •
2.4. The Type D Critical Region:
Balanced Design. • • • • • • • • • • • • •
2.5. The Type D Critical Region for a Size-a
Combination Procedure in the Two-Sample Case
20
26
28
29
35
37
2.5.1. Case 1.
2.5.2. Case 2••
2.5.3. Case 3.
2.6. The s-Sample Case • • • • • •
2.7. A Note on Applications.
iii
\
\,
....
39
40
Page
III. FURTHER CONSIDERATIONS IN THE LOCALLY OPTIMAL
COMBINATION OF BETA-DISTRIBUTEp STATISTICS •
3.1. Introduction. • • •
3.2. The Type F Critical Region:
Unbalanced Design •
3.3. Bahadur Efficiency.
3.4. Extensions to Other Problems.
IV. APPROXIMATE CRITICAL VALUES FOR TYPE D
AND TYPE F TESTS • • • • • • • • • • •
42
42
42
46
55
60
V. TABLES OF CRITICAL VALUES FOR TYPE D
AND TYPE F TESTS AND A NUMERICAL
ILLUSTRATION • • • • • • • • •
65
5.1. Tables of Critical Values for
Type D and Type F Tests
5.2. Numerical Illustration.
80
VI. SUMMARY AND SOME SUGGESTIONS FOR
FURTHER RESEARCH • • • •
65
86
APPENDIX A:
USEFUL MATHEMATICAL RESULTS
88
BIBLIOGRAPHY
..............
95
iv
LIST OF TABLES
Table
Page
4.1. Values of the Kolmogorov-Smirnov Statistic for
Certain Simulation Problems, Assuming Certain
Parameters are Constant for All Samples • • • .
63
4.2. Values of the Kolmogorov-Smirnov Statistic for
Certain Simulation Problems Without Restriction
on the Constancy of Parameters • • • • • • • • •
64
5.1.1. Critical Values for Two-Sample Type D
and Type F Tests. •
5.1.2. Estimated Critical Values for Type D and
Type F Tests for Certain Interpolated and
Extrapolated Values of the Parameters
a , a , and f
.
2
1
5.2.1. Distribution of Patients Among Six
Hypertension Treatments in Two Clinics.
v
\
66
79
80
LIST OF FIGURES
Page
Figure
2.5.1. Type D Critical Regions for Combining Two
Beta-Distributed Statistics • • • • •
29
3.2.1. The Ellipse Formed by the Approximate
Intersection of the Power Surface and a
Horizontal Plane at a Height of a + E •
45
vi
CHAPTER I
INTRODUCTION
1.1. Statement of the Problem and
Outline of the Research
Let Ql, ••• ,Qs be s independent statistics for testing the same
null hypothesis against the same alternative.
The problem is to combine
the results of these s tests into a single overall test for the common
testing problem.
Most combination procedures available in the litera-
ture are based not on the statistics Q. but on their probability inte1.
gral transforms Pi' where Pi
value of Q for i
i
= Pr(Q.1 .>-
= l, ••• ,s.
q.1. IH0 ) whereq.1. is the observed
Those combination procedures which have
been based on the Q. 's are mainly limited to the case of univariate
1.
hypotheses with test statistics distributed in the one-parameter exponential family.
The aim of this dissertation is to extend these latter
results to certain multivariate and multiparametric testing problems.
It is assumed throughout that the distributions of the statistics are
continuous.
Let the hypothesis be
H:
o
A = _0
A
(1.1.1)
with alternative
A ., A
-
-0
Statistics used in the testing of this hypothesis include Hotelling's
2
T
statistics and statistics used in conjunction with the general linear
j
2
model.
If X =
~§
+ £, where
~
NXq design matrix,
~
is an NXm matrix of observations, A is an
is a qXm matrix of unknown parameters, and £ is an
NXm error matrix, then the general linear hypothesis is
R:
o
C~M =
0
with alternative
where
~
and
of zeros.
~
are appropriate gXq and mxu matrices and 0 is a gXu matrix
If g=p and u=l or g=l and u=p, then
tor, respectively.
specified above.
C~M
is a pXl or lXp vec-
In either case, the null hypothesis is of the form
Now assume that C and M are of full rank and let
- -- - - - -- E = M' X' [I-A(A' A) -lA'] XM.
- - - - --
R = M'X'A(A'A)-lC' [C(A'A)-\;;'J-lC(A'A)-lA'XM
and
- - - ---
Statistics commonly used in testing (1.1.1) are based on the characteristic roots t , i = l, ••• ,[min(g,u)] of Ig-t~l: Roy's largest root
i
2
(max t.); the Rotelling-Lawley T (~t.); and Wilks' ~ (TI(l+t.)-l). In
i
1
i
0
1
i
1
the current problem there is only one root, t, so that Roy's statistic
2
and T are identical.
o
Under the null hypothesis and assumptions of
normalitY,(N-q)t/g = F is an F-distributed variate with g and N-q
degrees of freedom.
The above model can, for the type of hypothesis
being considered, be restated as R :
o
where a is a non-null vector.
-
C~Ma
----
:f 0,
Then the common test statistic is
F(a) = (N-q)/g[a'Ra/a'Ea], which has an F distribution with g and N-q
- -- -
degrees of freedom.
--
(Anderson [1958], Morrison [1967]).
Since the 'F distribution is difficult to work with, it is
assumed that the statistics have been transformed to beta-distributed
random variables.
If F has an F distribution with p and q degrees of
3
freedom, then U
p/2 and q/2.
=
(pF/q)/(l+pF/q) has a beta distribution with parameters
If F is noncentrally distributed with parameter 8 t U has a
noncentral beta distribution with the same noncentrality parameter.
If
F has a noncentral F distribution with noncentrality parameter 8 and
degrees of freedom p and q, then the density of F is given by
-8/2
~
00
1:
r=O
q
8r (pf/q)p/2+r-l
2r rlS(p/2+r t q/2)(1+pf/q)(P+q)/2+r '
0,
f > 0;
f <
o.
When 8=0, the density reduces to that of a central F distribution,
given by
(p/q)p/2 f P/ 2- l /[S(p/2,q/2)
for positive values of f.
(l+pf/q) (P+q)/2]
If F has a noncentral distribution and U is
as defined above, then U has a noncentral beta distribution with noncentrality parameter 8 and parameters p/2 and q/2.
The density of U is
given by
00
e- 8 / 2 E
o 2. u 2
0,
otherwise.
r=O
1;
(1.1. 2)
When 8=0, U has a central beta distribution and density
(1.1.3)
O<u<l.
In this dissertation some locally most powerful combinations of
beta-distributed statistics used in tests of (1.1.1) are developed.
The
concepts and notions of type D and type F tests, which maximize different measures of average curvature of the power surface at A , are
.... 0
4
presented in Section 1.3.
In Chapter II the type D test is determined for designs which
are balanced as defined in Section 2.3, with details for the two-sample
case.
In Chapter III the type F test, which can be applied in the
case of an unbalanced design, is determined.
It is also shown that if
there are a large number of finite-sample statistics to be combined, the
test based on the type D region is more efficient than Fisher's test
when the comparison is made on the basis of Bahadur asymptotic relative
efficiency.
Furthermore, it is shown that the type F test is more
efficient than Fisher's test when the comparison is made on the basis
of a measure of average Bahadur asymptotic relative efficiency.
A sec-
tion mentioning some extensions concludes this chapter.
The two-sample results discussed in Chapter II are difficult to
extend to cases involving even a moderately large number of statistics
to be combined.
However, the overall statistic is a linear combination
of the independent statistics to be combined, and the Central Limit
Theorem does apply.
Finite-sample approximations to this asymptotic
result are studied in Chapter IV.
Chapter V includes an example of applications of the theory to
some clinical data.
Some tables of critical values for the two-sample
case are also presented.
A final summary chapter concludes the disserta-
tion.
1.2. Review of the Literature
Recall that the problem is to combine the results of s independent tests into a single overall test for the common testing problem.
5
Let Qi' i
= l, ••• ,s,
be the test statistics and let Pi' i
=
l, ••• ,s, be
their probability integral transforms, as defined in the preceding
section.
Many of the combination procedures in the literature are
based only on the Pi's.
x=
Fisher [1932, pp. 99-101] proposed
-2 log Pl··· Ps
as an overall test statistic.
Under the null hypothesis, X is
distributed as a chi-squared random variable with 2s degrees of freedom.
Independently of Fisher, K. Pearson [1933] proposed
as an appropriate statistic.
Further discussion of these methods is
given by E. S. Pearson [1938], Wallis [1942] and Oosterhoff [1969].
Tippett [1931] proposed that the null hypothesis be rejected
if the smallest of the P. 's is sufficiently small.
1.
Wilkinson [1951]
later extended this test, suggesting that the k-th smallest of the P. 's
1.
be used as the statistic for the overall test, where k is a predetermined value.
Liptak [1958] proposed combinations of the form
where
-1
~
is the inverse of an arbitrary function ¢ and the w. are
suitable weights.
1.
Liptak suggested that one natural choice of ¢ is the
normal distribution function.
function, so that ¢
-1
If
~
is taken to be the exponential
is the logarithmic function, this test reduces to
a weighted version of Fisher's test, proposed earlier by Good [1955] and
applied by Zelen and Joel [1959] in the case where the Q. are distrib1.
uted according to an F distribution and s
= 2.
Lancaster [1961] also
6
discussed adding weights to Fisher's method.
Birnbaum [1954] compared the methods of Fisher, Pearson, and
Wilkinson.
lems.
He found that no method is optimal for all combination prob-
However, he showed that if the test statistics have distributions
belonging to the exponential class, then Pearson's method and Wilkinson's method with k>2 are inadmissible.
discussed in Section 2.2.)
(Admissibility is defined and
He also noted that while Tippett's method
is admissible, Fisher's method seems to be somewhat more uniformly
sensitive to commonly chosen alternatives.
Naik
[1969] compared Fisher's method to Tippett's method, and
Wilk and Shapiro [1968] compared Fisher's method to the mean of the
normal transform for empirical studies.
In both papers it was noted
that Fisher's method is not necessarily optimal for all testing problems.
Littell and Folks [1971] compared Fisher's method, Tippett's
method, Wilkinson's method with k=s, and Liptak's method, with ¢ defined
as the normal distribution function, using Bahadur relative efficiency.
Assuming that there were a fixed number of large-sample statistics to
be combined, the authors showed that Fisher's method is generally at
least as efficient as the other methods.
However, for some testing
problems, not all methods can be distinguished on the basis of Bahadur
efficiency and in these cases other methods may be superior.
In a later
paper, Littell and Folks [1973] showed that, under the same assumption,
Fisher's method is at least as good as any other method on the basis of
Bahadur relative efficiency.
The problem of finding optimal combination procedures is still
open, however, since it is often the case that researchers have sample
7
sizes insufficient to satisfy the asymptotic arguments of Bahadur
efficiency.
Furthermore, as noted by Littell and Folks, Bahadur effi-
ciency cannot distinguish Fisher's method from some other methods for
certain testing problems.
All of the tests discussed thus far are based on the probability
integral transforms of the s test statistics which are to be combined.
Various justifications for using the transforms rather than the test
statistics themselves have been proposed.
Birnbaum suggested that
(i) the values of Ql, ••• ,Qs may not be known to the statistician,
(ii) the distributions of the Q., which are not necessarily the same for
1
all i, may not be known to the statistician, (iii) although the values
and distributions of the statistics may be known, there may be no
available overall test based on the values of the statistics themselves,
or (iv) available methods may be difficult to apply.
Usually, however,
the statistician does have the relevant information concerning the
statistics to be combined.
If the available methods which are based on
the values of the statistics themselves are difficult to apply, the
statistician may prefer a simpler method based on the probability
integral transforms if the loss in power is small.
The arguments that
combination procedures which do utilize the values of the statistics do
not exist may be losing its validity as research in this area proceeds.
Van Zwet and Oosterhoff [1967] and Oosterhoff [1969] proposed
procedures for combining Ql and Q which are based on the statistics
2
themselves.
These authors considered the cases when the distributions
are either normal. or Student's-t distributions, but noted that their
techniques could be applied to derive results if the statistics have
distributions belonging to the one-parameter exponential family of
8
distributions.
They proposed three overall tests and compared them with
Fisher's test, finding none of the four procedures to be uniformly
superior for the problems considered.
Van Elteren [1958] considered linear combinations of two-sample
Wilcoxon tests, giving attention to the properties of tests based on two
different sets of weights.
Bhattacharya [1961] compared three procedures for combining
independent chi-squared statistics for sampling experiments.
He found
Tippett's method to be, in general, inferior to Fisher's method and to
the overall test based on the addition of the s independent chi-squared
statistics, but no uniform preference between the latter two methods
was observed.
The additi~n of independent chi-squared statistics for
contingency table tests to obtain a test over several control groups is
a well known procedure (see, for example, Hill [1971]).
Mantel [1963]
has proposed extensions of the Mantel-Haenszel procedure to various
related contingency table problems.
Schaafsma [1968] studied certain aspects of linear combinations
of normally distributed statistics.
Oosterhoff [1969] reported that
Davies also considered procedures to combine normally distributed
statistics, investigating B-optimal size-a tests to test the variance
of the normal distribution.
In the papers mentioned above, it has generally been assumed
that the statistics to be combined have continuous distributions.
In
this dissertation consideration will be limited to the continuous case.
Aspects of the discontinuous case have been discussed by Wallis [1942],
Lancaster [1949], E. S. Pearson [1950], and Kincaid [1962].
9
1.3. Introduction to Type DandType F Critical Regions
As mentioned earlier, for certain multivariate testing problems
type D and type F critical regions provide two locally most powerful
tests found by maximizing different measures of the curvature of the
power surface at _0
A.
In this section the preliminary notions concerning
the type D and type F critical regions are presented.
1.3.1. Type D Critical Regions
Uniformly most powerful tests do not exist for many testing
problems, but often locally most powerful tests can be found.
For
A ~ Ao , where A is a one-dimensional
testing H :
o
parameter, the locally best unbiased critical region has been described
by Neyman and Pearson [1936].
This so-called type A region of an
unbiased test is found by maximiZing the curvature of the power
A = Ao , for a fixed size of the test.
at the point
A = -0
A
Generalizing to a multiparametric testing problem, H:
o
versus HI:
curve
A
-
~
A where
-0
~
....
is a p-dimensional vector, it is reasonable
to attempt to find the locally best unbiased region by maximizing the
curvature of the surface of the power function at the point A = A •
-
-0
Among the class of all unbiased tests of a given size a, the test which
has the critical region found by simultaneously maximiZing the curvature
at every cross-section at the point
A = _0
A does not exist.
_
Pearson [1938] proposed a type C region, for the case p = 2.
Neyman and
A type C
region for a testing problem is defined to be the region with the best
local power along a given family of concentric ellipses with the same
shape and orientation.
If the concentric ellipses are circles, then the
type C region is said to be regular; otherwise it is nonregular.
How-
ever, to construct a type C region it is necessary to know the relative
10
importance locally of type II errors, and this is not usually known.
Furthermore, type C regions are not
in~ariant
which are one-to-one and twice differentiable.
under transformations
Therefore, transforma-
tions of regular regions may be nonregular.
To overcome these objections, Isaacson [1951] proposed a type D
region, which is defined as the unbiased critical region of size a which
maximizes the Gaussian curvature of the power surface at the point
A = A.
-0
The type D region can be constructed without knowledge of the
type II errors and is invariant under one-to-one, twice differentiable
transformations.
Geometrically, a type D critical region provides
intuitively appealing justifications for its adoption as an extension
of a type A critical region.
First, the type A critical region maxi-
mizes the curvature of the power curve at the point specified in the
null hypothesis and the type D region maximizes an average measure of
the curvature of the power surface at the point specified in the null
hypothesis.
Also, since a type A region maximizes the curvature of
the power function in an infinitesimal neighborhood of A = A , the
o
horizontal chord intersecting the power curve at a height a + £
is
minimized among all chords so constructed for unbiased tests of size a.
Similarly, among all unbiased critical regions of size a, the type D
critical region minimizes the area of the region on a horizontal plane
at a height of a +
£
which is defined by the intersection of the plane
and the power surface.
For simplicity, the definition of a type D critical region will
be presented in the two-parameter case.
number of parameters is immediate.
The extension to any finite
Adapting the notation of Isaacson,
for any region w in the set of all possible critical regions for a test,
11
the power function is given by
B(~lw)
where
E=
= Pr(U
E
wl~)
(U 'U ) is a sample point in a 2-dimensional sample space.
1 2
Here, each U is distributed as a beta with parameters b and a and noni
centrality
H :
o
e = f(A-A
)
.. ..0
for some function f.
A = ..A0 is equivalent to testing H:
0
Notice that testing
= 0,
A
A
..0
in each instance is that equality does not hold.
if the alternative
If the criteria used
in determining a type D region are invariant under translation, there
= ..O.
is no loss in generality in assuming that ..0
A
Thus the size of the
test associated with the critical region w is given by
B(A..0 Iw)
= B«O,O)' Iw) = Pr(U..
E
wl(o,O)') •
Now define
Bi = Bi(w)
= a[B«A l ·A 2), Iw)]
a Ai
i
(A ,A )
1 2
=
assuming that the derivatives exist.
1,2.
(0.0)
a2 [B«Al,A Z)' Iw)]
a Ai a Aj
=
i,j = 1,Z,
(Al,A )
Z
= (0,0)
A necessary condition for unbiased-
ness is that
(1. 3 .1.1)
The Gaussian curvature of the power surface at A
=
° is given by
= det(B..w) •
the determinant of B , where
..w
12
B
...w
=
[B ll (w)
B (w)
12
The restriction (1.3.1.1) guarantees that a relative extremum exists at
A
=~.
In the two-dimensional case, the extremum is a minimum if the
Gaussian curvature is positive and if the curvature in all directions
is positive, i.e., if det(B ) > 0 and B.. > 0, i
...w
J.J.
~
1,2 (Taylor and
~funn
[1972, p. 227]). Notice that these two restrictions are necessary and
sufficient conditions that B be positive definite.
...w
=
ditions that B
l
is unbiased.
B
2
=
0 and
~w
Together, the con-
is positive definite imply that the test
The type D critical region maximizes the Gaussian curva-
ture, det(B ), among all regions w which provide unbiased tests of a
...w
specified size.
Since the Gaussian curvature is invariant under trans-
lation, the above translation of ...0
A to 0 is not restrictive.
To sum-
marize:
Definition 1.3.1.1. (Isaacson (1951]).
A region w
o
is said to be an
unbiased critical region of type D for testing the null hypothesis that
A = 0 if:
(i) B(Olw )
...
0
= a;
(1. 3.1. 2)
= 1,2;
(1.3.1.3)
(iii) B
is positive definite;
...w
(1.3.1.4)
(ii) B. (w ) = 0,
i
0
J.
o
(iv) det(B
...w
o
) > det(B ) for any other region w
-
...w
(1.3.1.5)
satisfying (i) - (iii).
Returning to the discussion of the geometrical interpretation,
consider the power surfaces of size-a unbiased tests of a hypothesis Ho
13
and the set of regions on a horizontal plane at a he.ight of
Ct
+
E
which
are defined by the intersection of the horizontal plane and the power
surface.
It will be shown that' the region which is formed by the inter-
section of the plane and the power surface of the test based on the
type D critical region has minliual area among all regions in the set.
Consider the Taylor series expansion of the power function around O.
Ignore terms of third and higher orders and assume that the derivatives
of third order exist.
Then
If Al and A are in an infinitesimal neighborhood of
Z
excluded flom the expansion are negligible.
definite, E
=
~o
=
~A'B
ellipse defined by 0
TIO/ (det(~w)'
mized.
A > O.
- -w-
~,
the terms
Since B is positive
-w
For a fixed value of E, the area of the
= ~'~w~ is given by ,fO/
h Il B22 -Bf2
=
For fixed E, this area is minimized when,det (~w) is maxi-
By (1.3.1.5), w is the critical region which maximizes det(B )
o
-w
and thus minimizes the area of the ellipse.
To extend the definition to any arbitrary number, p, of
p~rameters,
define
~
and 0 as pXl vectors, increase the range of the
index i in (1.3.1.3) to p, and recognize Band B
as pXp matrices.
-w
-wo
The only point which requires further explanation is the verification
that B
-w
being positive definite assures that a minimum
exist~
at the
o
extremum point A
-0
O.
A pXp matrix _w
B
only if, for every nonzero p-vector
is positive definite if and
o
~, ~'~w ~
o
> O.
Assume that A
0
14
is not a point minimum of the power surface.
'\ *,
A
Then there exists a point
* ••• ').p)
* in an infinitesimal neighborhood of
().l'
=
*
° such that
B(). Iw ) < B(Olw ).
-
0
-
(1. 3. 1. 6)
0
By a Taylor series expansion, ignoring terms of third and higher orders
and assuming that the third order derivatives exist,
B().*lw)
-
0
= B(Olw
) +
0
0 +
* p
~(E~~= 1 ).~B
.. (w ) A.. + 2E
~ ~~
0
~
.. 1
~,J=
* .. (w)A..)
*
A..B
~
~J
J
0
i<j
< B(Olw )
-
0
by (1. 3.1. 3) and by assump'tion (1. 3.1. 6).
The inequality in the last
two lines implies that A. * 'B A.* < 0, which is not the case since B
is
-w0
-w0 by assumption positive definite. Thus the assumption that BO * Iw )
-
does not attain a minimum at the point
desired conclusion.
=~
~
0
is false, which is the
Therefore, B being positive definite is suf-wo
ficient to show that the unbiased test achieves a minimum at A.
O.
1.3.2. Type F Critical Regions
The type F test is based on the concept of best average power,
defined by Wa1d [1943] as follows:
Definition 1.3.2.1.
H:
o
).
=0
A critical region w for a testing problem
versus HI:
o
~ ~ ~
is said to have the uniformly best average
power with respect to a family of surfaces R (A.) and a weight function
-
6(~),
c -
if,
for any other critical region w of the same size,
f
R ().)
c ...
Pr(w IA.) 6(A.) dA > f
Pr(wlA.) 6(A.) dA
0 R ().)
c -
15
where f
$(A) dA is the surface integral of $(A) over R (A).
_
Rc (A)
If the surface is a sphere of radius r, so that A'A
function
c -
-
~(A)
-
= r 2 , and the
is a constant for all A, then it can be shown (in
-
Weyl [1939] and more simply in Roy
~
al. [1971, p. 25]) that
This implies that if a deviation from the null hypothesis is equally
likely in any direction, given
~'~ = r 2 , so that there is a uniform
distribution over the points on the p-dimensional sphere of radius r
with center at the origin, then the expected value of A'B A, the power
- -wof the test at _A, is proportional to the trace of -w
B. Since the trace
of a positive definite matrix is positive, to maximize the expected or
average power of the test subject to the restriction A'A
only to maximize tr(B
) with respect to
_w
w~
= r 2 , one needs
Following the notation of
Neyman and Pearson and Isaacson, a critical region which provides the
test with the best average local power will be termed a type F critical
region.
(Isaacson [1951] extended the notion of a type D critical
region to include nuisance parameters and called such a region a type E
To summarize, the following definition is offered:
region. )
Definition 1.3.2.2.
A region w
o
region of type F for testing H :
o
is said to be an unbiased critical
A
=~
against HI:
A:fOifw
o
satisfies
(i) - (iii) of Definition 1.3.1.1 and if
) > tr(B ) for every other critical region w
-w
o
satisfying (i) - (iii).
(iv) tr(B
_w
16
Although both the type D and the type F tests provide tests
which are locally most powerful on the average, by some means of
averaging, to be consistent with Wald's terminology, a type D test will
be referred to as a locally most powerful test while a type F test will
be termed a test which is locally most powerful on the average.
I .
CHAPTER II
TYPED CRITICAL REGIONS FOR COMBINING
BETA-DISTRIBUTED STATISTICS
2.1. Introduction
In Chapter I a critical region w was defined as an unbiased
o
critical region of type D for testing the null hypothesis that A = 0 if
w satisfied four conditions.
o
In this chapter the concept of permissi-
bility of a test is introduced and the form of the critical region of
permissible tests is considered.
By use of the concept of permissi-
bility, it is next shown that a type D critical region does exist for
the test procedure combining s independent statistics, U , ••• ,U ' where
1
s
U has a beta distribution.
i
The exact form of the type D critical
region is found under the assumption that the design is balanced in the
sense that the noncentra1ity parameters are proportional with known
proportionality constants.
Under this assumption, expressions for find-
ing critical values of size-a type D tests for combining two statistics
are given.
Finally, applications of these results to some F-distributed
statistics which arise in analysis are mentioned.
2.2. Permissible Tests and Critical Regions
There are two conditions which any reasonable test which combines statistics must satisfy.
The first is that the combining test
should depend on all of the statistics to be combined.
The second
condition, given by Birnbaum [1954, p. 562], requires that for every
18
point in the critical region of the test, every other point which
represents a more extreme deviation
be in the critical region.
fr~m
the null hypothesis must also
The term "permissible test" is introduced
to mean any test which satisfies these two conditions and is formally
defined as follows:
Definition 2.2.1.
Let U., 1
1.
~
i
~
s, be s independent statistics for
testing a null hypothesis H against an alternative HI.
o
Let W be the
sample space, a subset of s-dimensional Euclidean space.
If H is
o
rejected for large values of U., then a test of size a, 0 < a < 1, for
1.
combining Ul, ••• ,U ' with its corresponding critical region w, is pers
missible if:
(i) w depends on each D., 1 < i _< s, in the sense that
1.
-
Pr(U = (Ul' ••• ,U., ••• ,U )
-
1
S
E
wlu.; U
1_
E
W) is not constant
for all values of U.; and
1.
(ii) for any arbitrary point (ul' ••• 'u ) in w, every point in
s
the set
u! >
1 -
U.,
1
1 < i
< s;
U
'EW} (2.2.1)
is also in w.
If H is rejected for small values of U., then the definition holds when
o
1.
the inequality in (2.2.1) is reversed.
Since any reasonable combination procedure is permissible, permissibility cannot be used to distinguish effectively between competing procedures.
Birnbaum noted, however, that more restrictive conditions
which would narrow the number of choices cannot be specified without
further assumptions.
Having assumed that the densities of the
19
statistics to be combined all belong to the one-parameter exponential
family, he considered properties of the acceptance regions of admissible
tests, where admissibility is defined as follows:
Definition 2.2.2.
(Birnbaum [1954]).
A test of a hypothesis is
admissible i f there is no test of that hypothesis with the same level
of significance which, without being less sensitive to any possible
alternative hypothesis, is more sensitive to at least one alternative.
Birnbaum [1955] showed that if the densities of the statistics belong
to the exponential family, a necessary condition for the admissibility
of the test is that the acceptance region be convex.
Since the non-
central beta distribution is not a member of the one-parameter exponential family with respect to the noncentrality parameter, Birnbaum's
conclusion on the convexity of the acceptance region does not apply.
However, the following characterization of permissible critical regions
does apply and will be useful in subsequent derivations:
Theorem 2.2.1.
Let Ul, ••• ,U
s
be s independent statistics such that each
U is distributed continuously over a finite or an infinite interval I •
j
j
Let w be a size a, 0 < a < 1, critical region for a test
~(U)
and assume that this test is
permissi~le.
~(Ul' ••
"Us)
Also assume that the
boundary between the rejection and acceptance regions is included in
the rejection region.
Then for any i, 1
~
i
~
s, the boundary between
the acceptance and rejection regions can be written
1 ~ j ~ s,
where f
i
is a function which, in each of its (s-l) arguments, is non-
increasing and is strictly decreasing at at least one point.
20
Proof.
=
If a
0 or a = 1, the f
are not necessarily as specified;
i
l~
hence the restriction that 0 <a <
Also, since the U.1. are continuous,
.
there is no loss in generality in assuming that the boundary is included
in the rejection region.
Since the test is permissible, it is clear from Definition
~(l) =
2.2.1 part (ii) that if, for any j,
u(l)
E
>
u
U'
j
Wand u(2)
= (ul""'u~, ••. ,u
s
J
)
E
(ul, •.• ,uj, •.. ,u )
s
4 w but
w, it must be the case that
r
First let i
= 1. Now assume that the boundary between the
acceptance and rejection regions is increasing with respect to some
argument, say u •
2
~(3)
E
W, and u(4)
=
(ul,uZ, ... ,u )
s
w where
E
U
z < u z.
In this instance
the test is not permissible; therefore, the boundary cannot increase
By the same argument, the boundary does not increase in u.,
J
Next assume that the boundary does not depend on u. for
3 ~ j ~ s.
some j, 2
J
~
j
~
s.
Then part (i) of Definition Z.2.l is not satisfied.
Therefore, the boundary must depend on each u ' 2
j
boundary depends on each
2
~
U.,
J
2
~
j
~
Since the
J
j
can be repeated for i
j < s.
s and is nonincreasing in each u.,
j ~ s, it must decrease in each u ' 2 ~ j
This completes the proof if i
~
= 1.
< s.
Since the above argument
= Z, ••• ,S, the theorem holds.
2.3. The Existence of a Type D Critical
Region for Combining s Independent
Statistics with Beta Distributions
In this section the following theorem is proved:
Theorem 2.3.1.
Assume that U , 1
i
~
i
~
s, are s independent statistics
and that each U has a beta distribution with parameters band a
i
i
and
21
noncentrality parameter e •
i
eters are of the form e
i
Also assume that the noncentrality param-
= A'ViA
-
-
where each V., 1
-'
-1
positive definite matrix and A is a pXI vector.
~
i
~
s, is a pxp
If each U. is a
1
statistic for the testing problem defined by (1.1.1), then an unbiased
critical region of type D does exist for the overall test of H which
o
Proof.
To verify the existence of a type D critical region for the
particular problem specified in the theorem, it must be shown that the
four conditions (1.3.1.2) - (1.3.1.5) hold for this problem.
condition specifies the size of the test.
o -<
The first
Let
u.1 <
1, 1 < i ~ s}
-
(2.3.1)
and let w be a critical region, w c W, such that B(9!w) = a.
Since the
densities of the U. are continuous in W, this condition can be satisfied
1
for any predetermined value of a, 0 < a < 1.
Once the form of the
critical region is known, its exact boundary can be found for every
such a.
The first condition can thus be satisfied.
To verify (1.3.1.3) for the particular problem of this theorem
it must be shown that
s
eh
Bi(w) = [a f IT (Su duh)/aA i ]
w h=l
h
A=O
I
--
s
eh
= {Ljs=I[a f IT (Su duh)/ae. x cae./aA')]}1
w h=l
h
J
J
1
where
eh
Su
theorem.
h
0, 1
~
i ~ p,
A=O
(2.3.2)
is defined in (1.1.2) with e
h
of the form specified in the
For each i = I, ••• ,p and j = I, ••• ,s,
22
oOj/OA
t
act
o(A'V.A)/oA =
i
- -J-
i
m'=l m=l
A A , V(j»/dA
m m mm'
i
A vq)
= 2 [p
m=l m m~ ,
(2.3.3)
where v(j) = v(j) is the mm'-th element of V.•
m'm
nun'
~J
For any i and j, (2.3.3)
is zero if A = O.
B.
Therefore, (2.3.2) holds and
~
(w) = 0 for all
i = l, ••• ,p for all critical regions w.
The third condition to be satisfied requires that the matrix
« [a2B (A- Iw) / (d A. dA.1 ,) ] I
B =
-w
))
~
A=O
be positive definite.
In the problem being considered here, the ii'-th
element of B is
-w
2
s
8h
[d {! II (B
duh)}/(dA.dA")]1
1
1
w h=l U h
8h
Bu
By expanding the densities
(2.3.4)
•
A=O
.
in their series representations given
h
by (1.1.2), it can be shown (see Appendix A, Lemma A.I) that (2.3.4)
equals
r 02
~'aA OA.
L
i
00
-8 /2
00
[E ... E
l' r
1
e
1
=0 r =0
r
(8 /2) 1/(r ! BCb+rl,a »
l
l
1
s
-8 /2
x ••• xe
x!
s
II
s
r
(8 /2)'
s
s
b+r -1
(u.
j
J
w j=l
a -1
(l-u.) j
J
}
du.)]
J
I
A=O
2
a
= aA i dA i'
/(r ! B(b+r ,a »
s
s s
g[w;b;(8.,r.,a.), 1
1
~
1
~ ~ s]l,
J A=O
(2.3.5)
i
(2.3.6)
23
= g[wjbj(8 i ,r.,a.),
1
1
where g
in (2.3.5).
1 < i < s] is the term in square brackets
--
C2.~.6)
Upon differentiating,
[
t
~
J ~=~ +
°rs~x~
d)
d8.
~ j=l d8 j
dA i '
becomes
<;,s.
~
"
1 '"'8
J=
0
j
[
~
X
---L-)]
"" '" [d8.
""
_0_
• (2.3.7)
ol\i Ol\i'
;\=0
From (2.3.3), the first term of (2.3.7) is zero.
Also from (2.3.3),
(2.3.8)
It remains to evaluate [dg/d8 j ]
I '1~
j
< s.
Ignoring for the
A=O
moment the integral, consider
r
b+r.-1
a -1
e- 8j /2(8./2) j u. J
(l-u.) j
00
_d_
J
r
J
J
r .! S(b+r . , a .)
a8 j r.=O
J
J
~
J
-8./2
e
00
= r
J
(-1/2)
rj=O
r
r.
b+r.-1
(8./2) J u.
J
a-1
(l-u.) j
..::..J~~~J_~_---=.J
__
r.-1
-8./2
e
j
-A=O-
r j ! S(b+rj,a j )
00
+ r
J
=1
J
(r./2)(8./2) J
J
J
b+r -1
a -1
u. j
(l-u.) j
J
J
A=O
a.+b
= [-1/2 +
1/2(---,\-) u j ] S~.
(2.3.9)
J
since
[(a.+b)/b]/SCb,a )
j
J
= 1/S(b+1,a.).
J
In light of (2.3.5) - (2.3.9), (2.3.4) can be rewritten
(2.3.10)
24
s
(j)
Ej=l vii'
where ¢. = -1 + (a.+b)u./b.
J
J
J
s
0
h~l ~SUh
du h )
s
IT
(So du )
¢j
h
h=l
uh
w
v. J
~J
v.
~
¢j
Therefore
s
B = E. 1
-w
J=
=
s
~
x C (w)
(2.3.11)
j
j=l -J
s
where C.(w) = J ¢. IT (So du ).
h
J
w J h=l
uh
In order to show that _w
B is positive definite, it suffices to
show that the coefficients CJ.(w) > 0, 1
are by assumption positive definite.
tion to the two-sample case.
~
s, since the matrices V.
-J
Let j = 1 and restrict considera-
Then it can be shown that
Cl(w) = J SO
w
~ j
u1
SO
u2
¢1 dU
dU
1
2
whenever w is a permissible critical region.
> 0
By rewriting (2.3.11),
it can be seen that
=
(S
O
u
l
Su *)
0
l
dUll dU
2
(2.3.12)
where S0* denotes the density of a random variable U with a central
l
u
1
beta distribution with parameters (b+1) and aI' where [u2l denotes the
lower bound of u ' and where [u11u2] denotes the lower bound of u 1
2
given u •
Z
If w is permissible, then by Theorem 2.2.1, the region of
integration can be expressed in this manner as long as 0 < a < 1.
Assume that [u11u2]
= O.
Then the inner integral of (2.3.12) becomes
the integral over the range of positive density of the difference of two
25
density functions, which is zero.
Since a
l
+ b > b > 0, -1 + (al+b)ul/b
is a strictly increasing function of .u . which attains its minimum at
l
u
l
= Oif o ~
ul
It can, therefore, be concluded that
~ l-
1
I(u )
2
=
f
[ullu z ]
1
SO
u
cl>l dU l ~ f
0
l
1
SO
u
with strict equality only if [ulluz]
cl>l du 1
l
= O.
= f (Sou - Su0* )du l = 0
0
l
l
However, if 0 < ~ < 1 and w
is a permissible critical region, [ulluz] must exceed zero for values
U in some interval.
z
It thus follows that for any permissible critical
region w of size a, 0 < a < 1, I(u ) > O.
Z
Similarly CZ(w) > O.
Since SO
U
z
is positive in
Therefore, B is positive definite when s = 2.
-w
The demonstration that these coefficients are positive in the general
s-sample case follows in identical fashion.
Finally, it must be shown that there exists a size-a critical
region w such that
o
s
~ det[~1' __ 1
for all other size-a critical regions w.
form
s
~i=l ~i x
V.
x C.(w)] = det(B )
_1
1
-w
Since all matrices _w
B of the
Ci(w) are positive definite,
det(~w)
is a real positive-
values function of B •
-w
Also, B is not a constant for all w since
-w
Ci(w) is not constant.
It follows that det(B_w ) can be maximized and,
therefore, the final condition can be satisfied.
proof of Theorem Z.3.l.
This completes the
26
Although it has been shown that a type D critical region does
exist, it is not in general possible to find this region.
Since the
determinant function is not a finear operator, there is no simple method
of finding
(2.3.13)
In Section 2.4 it will be shown that the type D critical region can be
found if the design is balanced in the sense that V.
_1
where the y. are scalars.
1
= y.V,
1
1-
< i
~
s,
The case of an unbalanced design will be
considered in the third chapter.
2.4. The Type D Critical Region:
Balanced Design
In this section the following theorem, specifying the form of
the type D critical region in the case OC a balanced design, is proved:
Theorem 2.4.1.
Vi
-
= Y.V,
1-
If the conditions of Theorem 2.3.1 hold and if
1 < i < s, where the y. are known positive constants, then
-
-
1
the type D critical region of size a for the procedure combining
wo
= {(u1 , ••• ,u):
s
E1~-_1 Y.(a.+b)u./b
> k'; 0 < u. < 1, 1 _< i _< s},
1
1
1
1(2.4.1)
where k' depends on a.
Proof.
Then by (2.3.11), if
max[det(B )]
-w
w
= det(V)
P
x [max(E~1=1 y.1 C.(w»J
1
W
= det(V)
Yi
x [max(C )JP.
w
w
= YiY,
27
Since Y and Ci(w) are all positive, C is positive.
i
w
critical region is found by maximizing C.
w
Thus the type D
This maximum can be found
by applying the Neyman-Pearson Fundamental Lemma, stated here without
proof.
Lemma 2.4.1.
(The Neyman-Pearson Fundamental Lanma).
Suppose m + 1
given integrable functions g o ,gl, ••• ,gm are defined in an s-dimensional
space.
Consider the set of all regions w for which the following hold:
1 gi
W
s
IT
. 1
J=
dU j = c.,
i
= l, ••• ,m
(2.4.2)
1
where the c. are m given constants.
If w is a region which satisfies
1
0
(2.4.2) and if
m
~ E:=l
k.g.
1 1
g0 < L.1=1
k.g.
1 1
{ go
in w '
o
(2.4.3)
outside w '
o
for m chosen constants k , then w has the property that
0
i
s
s
1 go IT duo 2-1 go IT duo
J
J
j=l
w
j=l
w
0
for any region w which satisfies (2.4.2). (Isaacson [1951]).
s
0
s
s
0
Let m = 1, go = (IT S )(E'- l y. ¢.), and gl = IT S ,so that c l = a.
j=l U j
11 1
j=l u j
If k
l
=
(-L~=l Yi ) + k',
s
L -_ y.(a.+b)u./b > k'.
1
i l 1 1
then go
2- klg l if and only if
Since the reverse inequalities also hold
simultaneously, (2.4.3) holds for k
l
as defined above.
Therefore, the
type D critical region w , the critical region which maximizes C , is
o
w
as specified in the theorem.
This completes the proof.
28
2.5. The Type D Critical Region for a Size-a Combination
Procedure in the Two-Sample Case
In the previous section the form of the type D critical region
w was found.
o
Given a, the problem is to determine k' in (2.4.1) such
that
f
w
o
s
II
j=l
o
CSu.
J
du.)
J
= a.
In this section this problem is handled in the two-sample case.
For
simplicity replace k' by k = k'b/[Y2(a +b)] so that
2
where f
=
[y (a +b)]/[Y2(a +b)].
l
Z
l
The purpose is to find k such that
for a given a,
By
Taylor series expansions of the terms in the integrand, several
different expressions for the evaluated integral are found.
All of the
expressions involve double summations which have finite or infinite
numbers of terms depending on which, if any, of the parameters aI' a ,
2
and b are integers.
It is of practical interest to use the expression
which has the fewest infinite summations for a given set of parameters,
and it can be seen that each of the different expressions found is the
most useful in certain instances.
At the onset it is useful to notice that the type D critical
region is one of three different types, illustrated in Figure 2.5.1.
The boundary between the acceptance and rejection regions is a segment
of the line U
l
= (k-U 2)/f
or, equivalently, U
2
=k
- fUl'
29
For simplicity, in the following discussion let B.
1
let W be as defined in (2.3.1) with s
=
B(b,a.) and
1
= 2. Also, if x is a real posi-
tive number, denote
if x is an integer;
otherwise.
U
U
It
2
It ![\
\u1-(k-l) If
I
"
-_,u1-(k-l)/f
I
\'
::w··
•• 0
,
0
0
u -k-f
2
" ..... u
k'
I
II
0
0
k
I
'~
k
,
u
l
1
•••• w
••
0
•
•
-.:: u -k-f
2
'. u
o
o
I
~
f
'flL
::
.w"::
• •• 0 ..
k
l
.......
......
.....
o
u
1
1
Csse 3
Case 2
Case I
..
f
0
Fig. 2.5.1. Type D Critical Regions for Combining Two BetaDistributed Statistics
Again for simplicity, some of the lemmas in Appendix A are used
repeatedly without specific reference.
In particular, Lemma A.2 justi-
fies the interchanging of the order of iterated integration and
Lemma A.3 justifies interchanging the order of summation and integration
for this problem.
The expansions and results of other lemmas are also
used without reference.
Expressions relating k and a are now found for each of the three
types of area of integration.
From Figure 2.5.1 it can be seen that in Case 1, 0
o~
(k-l)/f < 1, 1
~
kIf, and 1 < k.
~
k-f < 1,
Under the assumption that w is
o
such that these conditions hold, an expression for (2.5.1) which is
30
useful when b is an integer is first found.
It is later shown that this
expression is also useful if none of the parameters, aI' a , and b, is
2
an integer.
Expressions which are more practical when aI' a , or both
2
are integers are also found.
First write
1 - 1
w-w
°
/30 /30 du du
u u
1 2
l 2
(k-l)/f
=1
- [1
o
1
/30 ( 1
0
ul
k-fu
-J
/30 (I
1 /30 du )du .
2
l
(k-l)/f u l 0
u2
1
+ 1
(2.5.1.1)
In the first term in brackets, the inner integral is unity, leaving
where I(k_l)/f(b,a ) is the incomplete beta function ratio.
l
available by Pearson [1968].
Vol. 2 [1970, pp. 48-51].)
evaluated by setting u
b-l
2
(Tables
For other references see Johnson and Kotz,
The remaining term in (2.5.1.1) can be
=
(1-(1-u
2
» b-1 and proceeding as follows:
31
1
= i3if"
1 2
(b-1) * (b-1)
E
i=O
.
1
l=!li
a 2+'1
(2.5.1.3)
The integral remaining to evaluate in (2.5.1.3) can be considered as
the difference of two integrals, the first of which is
(2.5.1.4)
Using Lemma A.6 the second integral can be evaluated as follows:
=f
=
a 2+i (b-1) * b 1
'
k-1 a 2+i+a 1+j
E
( : )(_l)J (1 - - )
13(a +j,a +i+1)
2
1
j=O
J
f
a +i
(l-k+f) 2
(2.5.1.5)
Since both (k-f) and (k-1)/f are between zero and one in Case 1, neither
(l-k+f) nor (1 - k~l) increases as its exponent increases.
Combine
(2.5.1.4) and (2.5.1.5) and apply Lemma A.5 to show that (2.5.1.3) is
equal to
32
(b-I)* b-I i=!li
a +i
1:
(.)
+' (I-k+f) Z
1
aZ 1
i=O
x
(b-I) *
1:
j=O
(2.5.1.6)
These series converge in light of Lemma A.7.
Substitute (2.5.1.2) and
(2.5.1.6) into (2.5.1.1) and cancel terms where possible.
Then i t is
seen that (2.5.1.1) can be written
(b-l) * b-I (_I)i
a 2+i
1:
( , ) +' (l-k+f)
i=O
1
a2 1
x
*
(b-l)
a +j
( bJ-.l) (-1) j (1 _ k-fl) 1
1:
S(a +j,a +i+l).
1
2
j=O
If b is not an integer but either a
l
or a
Z
(2.5.1.7)
or both are integers,
another expansion of (2.5.1) may have fewer terms and thus be more useful.
Consider again the final term in (2.5.1.1).
It will now be
evaluated by expanding the (l_u,)a i terms rather than the
1
before:
a -1
x
(2
i
.
)(_1)1
u~-l
1
terms as
i\
33
*
(a 2-l) a -1
L (2 )
1
= f3 l f3 2
i=O
i
(aI-I)
*
L
j=O
1
x !
(k- f u )b+i d u •
ub+j-l
l
l
l
(k-l)/f
(2.5.1.8)
Let y = fUl/k then apply Lemma A.5 to find that the remaining integral
in (2.5.1.8) equals
kb+i {(k/f)b+jr8f/k(b+j,b+i+l) - 8 _ / (b+j,b+i+l)]}.
l l k
(2.5.1.9)
where 8 (c,d) = 8(c,d) x I (c,d) is the incomplete beta function.
x
x
From
the proof of Lemma A.7 the term in square brackets is bounded and even
if the summation over j is infinite, that series converges.
since k
~
1 in Case 1, a
2
However,
must be integral to assure the convergence of
the series summed over i.
By combining (2.5.1.1), (2.5.1.2), (2.5.1.8),
and (2.5.1.9) and applying Lemma A.9, a second expansion for the integral
over the critical region is found:
1
Il_(k_l)/f(al,b) - 8 8
1 2
a 2-l
a -1
i
(2.) (-1) kb+i
i~O
1
b+i
~
x rf3 / k (b+j,b+i+l) - B1 _ 1 / k (b+j,b+i+l)] •
f
If a
2
gration of U
l
is not an integer but a
~nd
U
2
can be reversed.
l
(2.5.1.10)
is an integer, the order of inteEssentially the same steps are
34
integra~
performed to find that, for
.
1
Il_(k_f)(aZ,b) - B S
l 2
aI' (2.5.1.1) can be written
a -1
1
a 1-1 (_l)i
b+i
i:O ( i )b+i (kIf)
(a -1)
2
j:O
* aZ-l
(
j
j )(-1)
x [kb+j{Bl/k(b+j,b+i+l) - Bl_f/k(b+j,b+i+l)}l.
(2.5.1.11)
By Lemma A.7, the term in square brackets is bounded and the j-indexed
series converges.
If neither a
1
nor a
Z
is an integer, other expressions for
(2.5.1.lZ)
can be found.
when (b-1) *
They are, however, no improvement over using (2.5.1.7)
= co.
Three expressions, (2.5.1.7), (2.5.1.10), and (2.5.1.11), have
been found which, given aI' a 2 , b, and f, can be used to find the k such
that (Z.5.l.l2) equals some predetermined value a whenever w is of the
a
form considered in Case 1.
It is not, in general, possible to solve
for k in terms of a, so iterative solutions must be reLied on.
The results of this section are formalized in the following
theorem:
Theorem 2.5.1.1.
For each i = 1,2, let SO
u.
be the density of a random
1.
variable U which is distributed according to a central beta distribui
tion with parameters b and a .• Let w be a critical region, given by
1.
0
i
where f is a known constant.
If, for a given a
€
= 1,2}
(0,1), (2.5.1.12)
equals a and i f k':' 1 and kIf':' 1, then alternative evaluations of
(2.5.l~12)
are given by (2.5.1.7), (2.5.1.10) and (2.5.1.11).
If either
35
(i) b is an integer and at least one of (a ,a ) is not an integer, or
l 2
(ii) neither aI' a
Z
or b is an integer, then (Z.5.l.7) is the most use-
ful of the three expressions.
a
2
If b is not an integer but either a
l
or
is an integer, then (2.5.1.11) and (2.5.1.10) are more useful than
(2.5.1.7); expressions (2.5.1.10) and (2.5.1.11) both apply if a
are both integers.
l
and a
2
If all three parameters are integers, all of the
three expressions are applicable and all have a finite number of terms.
2.5.Z. Case Z
In Case 2 there are essentially two subcases, as illustrated in
Figure 2.5.1.
Only the first alternative will be considered here, since
the second subcase can be transformed into the first by resubscripting
and proceeding accordingly.
"Case Z" will now be used to refer specifi-
cally to the first subcase, where 0
~
(k-l)/f < 1, 0 < kIf
2
1, and
1 < k.
In this case
!
SO SO
w
l
0
u
U
(k-l) If
z
duldu Z = 1 - [!
0
kIf
+ !
(k-l)/f
SO !
u
1
SO
U
l 0
SO !
u
l 0
k-fu
1
SO
u
2
z
dUZdu l
duZdu l ]·
This expression is exactly (Z.5.l.l) except that kIf replaces 1 as the
upper limit of the outer integral in the last term.
The first term in
brackets was evaluated in the previous section, so it remains only to
evaluate the second term in brackets and combine terms.
Since the
techniques of integration used do not differ from those in the previous
section, the details are omitted.
36
Corresponding to (2.5.1.7) is the expression
1
(b-l) * b-l (_I)i
a 2+i
Il_k/f(al' b) + - S
- r
(i)
-; (l-k+f)
S1 2 i=O
a 2+1
In Case 2, 0
~
(2.5.2.1)
x 8 (I-k+f) (a 2+i+l, a 1+j)·
1/
a +j
1
(k-l)/f < 1 so (1 -(k-l)/f)
does not increase with j.
However, kif
~
1 so 1
~
(l-k+f) and thus (l-k+f) may increase with i.
By Lemma A.a the i-indexed series converges anyway.
Corresponding to
(2.5.1.10) and (2.5.1.11) respectively are
1
I 1 _(k_1)/f(a 1 ,b) - Sl S2
(a -1)
2
i:O
*
(
a 2-1 (_l)i
i )b+i
b+'
k
1
(aI-I) *a -1
r (1 ) (-l)j(k/f)b+j ~1/k(b+i+1,b+j)
x
j=O
(2.5.2.2)
j
and
(aI-I)
r
*a
i=O
.
-1
1
.
(1. )(-:) (k/f)b+1
1 b+1
x
(2.5.2.3)
Lemmas A.a and A.7 can be used to show the convergence of these series
when kif
~
1 and k > 1.
To summarize:
Theorem 2.5.2.1.
Let U , SO , and w be defined as in Theorem 2.5.1.1.
i
u
0
If, for a given ex.
€
i
k
~
(0,1), (2.5.1.12) equals ex and i f 0
~
kif < 1 and
1, then alternative evaluations for (2.5.1.12) are given by (2.5.2.1),
(2.5.2.2) and (2.5.2.3).
None of these expressions restricts any of the
37
parameters to integral values, although (2.5.2.1) has the fewest terms
if b is an integer and at least one of a
(2.5.1.2) has the fewest terms if only a
has the fewest terms if only a
1
1
and a
2
2
is not an integer,
is an integer and (2.5.1.3)
is an integer.
As before, iterative solutions of k must be used since k cannot be found
explicitly for a given a.
2.5.3. Case 3
In Case 3, 0
~
k
~
1 and 0
~
kif
~
1.
Also in this case, the
integral in (2.5.1) can be written
I-kif
f
o
(k'-l)/f
f
o
where k' = (l-k+f).
(2.5.3.1)
If (2.5.3.1) and the term in square brackets in
(2.5.1.1) are compared it is seen that they are identical except that
38
(i) U.1 has a beta distribution with parameters . band a
beta distribution with parameters a
by k'.
i
i
while Y.1 has a
and band (ii) k has been replaced
Let
o 0
.
8 8
du du ::: I. [k;(b,a ),(b,a ),f]
2
u u
1
1
1 2
1 2
where Ii [k;(b,a ),(b,a ),f] denotes an expansion of the integral if
1
2
w ::: wei)
o
0
is a critical region which is covered in Case i, given the
parameters of the beta distributions and f.
Then the above comparison
of (2.5.3.1) and (2.5.1.1) reveals that
The appropriate substitutions in (2.5.1.7) and (2.5.1.10) give
rise to the folloWing expressions, which are valid if the critical
region is of the form considered in Case 3:
(a -1)*
1
1
- 8 B
1 2
2
~
i:::O
,
a 2-1 (_1)1 b+i
( i )b+' k
1
*
(aI-I) a -1
,
,
t
(1 )(_l)J(k/f)b+J 8(b+j,b+i+1);
x
x
b~l
j=O
x
(2.5.3.2)
j
j=O
(b-:-1) (-l)j(l _ k-1)a1+
J
j
f
~f/(1_k+f)(al+j.a2+i+l)
(2.5.3.3)
39
Since 0
~
k
~
1 and 0
~
~
kIf
1, neither a
2
nor a
l
in (2.5.3.2) is
restr,icted to integer values. ' However,. since (l-k+O and (1-(k-1)/f) may
exceed one, b is restricted to integer values in (2.5.3.3).
Although
(2.5.1.11) also has a corresponding expression in Case 3, it also
requires that b be integral.
What is needed is an expression which will
improve on (2.5.3.2) when aI' but not a , is an integer.
2
It is given by
(aI-I) *
1
1
1 2
L
- 8 8
i=O
(a 2-l)
x
*a
-1
(2
L
j=O
.
)(_l)J kb+j S(b+j,b+i+l).
(2.5.3.4)
j
The results of this section are now summarized:
a
Let U., S , and w be as defined in Theorem 2.5.1.1.
Theorem 2.5.3.1.
~
If, for a given a
o~
kIf
~
E
u
0
i
(0,1), (2.5.1.12) equals a and if 0
cable only if b is an integer.
l
1 and
1, then alternative expressions for (2.5.1.12) are given by
(2.5.3.2), (2.5.3.3), and (2.5.3.4).
that a
~ k~
and/or a
2
Expression (2.5.3.3) is appli-
Neither (2.5.3.2) nor (2.5.3.4) requires
be integral, but, for nonintegral b, if a
2
but not a
is an integer, then (2.5.3.2) has the fewest number of terms while if
a
l
but not a
2
is an integer, (2.5.3.4) has the fewest terms.
As before, k cannot in general be solved for explicitly in terms of a.
2.6. The s-Sample Case
In Section 2.5, discussion is limited to the two-sample case.
As the number of statistics to be combined increases, the number of
cases to be considered increases and the mathematics of evaluating
l
40
the integrals becomes increasingly cumbersome.
For small values of s,
say s = 3 and s = 4, it is probably not unreasonable to seek solutions,
although none are sought here.
As s increases much further, the labor
involved in evaluating the necessary integrals becomes extremely
tedious.
However, since the overall statistic for combining s independ-
ent statistics Ul"",Us is
where f
holds.
i
= yi(ai+b)/b,
it can be shown that the Central Limit Theorem
The approximate normality of Z will be studied in Chapter IV.
2.7. A Note on Applications
As noted in Chapter I, statistics with F distributions can be
transformed into beta-distributed random variables.
Therefore, the
results of this chapter can be applied to the statistics discussed in
Section 1.1.
If, for example, there are s independent Hote11ing's T
statistics to be combined, where (N
i
- p) T:/[(N.-I)pJ,i
].
].
2
= 1, ••• ,s,
has an F distribution with p and Ni-p degrees of freedom, then it
follows from the material in Section 1.1 that U.].
=
[T~/(N.-1)J/
].
l.
2
[l+T /(N -1)] has a beta distribution with parameters p/2 and (N -p)/2.
i
i
i
The results above can be applied by setting b
and calculating f accordingly.
= p/2
and a
i
=
(N -p)/2
i
In many actual experiments the number
of variates may be quite small in comparison to the sample size so that
b
=
[p/2] < [(N -p)/2]
i
= ai •
If this is the case and s
= 2,
then for
integral b the expressions derived in Section 2.5.1 - 2.5.3 which have
summations extending to (b-1) have fewer terms than the other
expressions.
In particular, if b = 1, then b - 1 = 0 and expressions
41
(2.5.1.7), (2.5.2.1), and (2.5.3.3) have only one term.
instance, k can be solved for explicitly in (2.5.1.7).
In this
Of course, for
other sets of parameters the other expressions presented in Sections
2.5.1 - 2.5.3 may have fewer terms and thus be preferable.
The results of this chapter also apply to the other statistics
mentioned in Section 1.1.
Again, if s = 2, the most useful expression
for finding k, given a, can be chosen on the basis of the parameters of
those statistics.
CHAPTER III
FURTHER CONSIDERATIONS IN THE LOCALLY OPTIMAL COMBINATION
OF BETA-DISTRIBUTED STATISTICS
3.1. Introduction
In this chapter the theoretical results of Chapter II are
extended.
In Section 3.2 a test which is locally most powerful on the
average is found in the case of an unbalanced design; the locally most
powerful test based on a type D critical region is in general not avai1able for this case.
In Section 3.3 it is shown that when there are a
large number of small-sample statistics to be combined, the locally most
powerful test found in Chapter II is more efficient than Fisher's test
according to Bahadur asymptotic relative efficiency, for local a1ternatives to the null hypothesis.
It is also shown that, for local a1terna-
tives, the type F test is more efficient than Fisher's test according
to an average measure of Bahadur asymptotic relative efficiency.
This
result is of particular interest in light of the work of Littell and
Folks [1971,1973] which was discussed in Chapter I.
In the final
section several other extensions of the results of the second chapter
are considered.
3.2. The Type F Critical Region:
Unbalanced Design
The case of an unbalanced design is considered in this section.
It is assumed that,under H , the s independent statistics uI, .•. ,U to
s
1
43
be combined have noncentra1 beta distributions with noncentra1ity
It is no longer assumed,
however, that the design matrices
~i
are proportional.
It is known
from Theorem 2.3.1 that a type D critical region does exist, but it is
not in general possible to find the type D critical region since it is
not in general possible to find
where Band Ci(w) are defined in (2.3.11).
-w
Although it is not always possible to find the type D critical
region in the case of an unbalanced design, it is possible to find the
type F critical region, which provides a test which is locally most
powerful on the average, as discussed in Chapter I.
From Definition
1.3.2.1 and (2.3.11) it follows that the type F critical region is
given by
s
max[tr(B)] = max[tr(L =l ~i x Ci(w»]
i
-w
w
w
=
s
max[L '=1 tr(V.) x C.(w)]
W
where Y
i
= tr(V.).
-1
-1
1
1
This maximum has already been found in Section 2.4
and the desired critical region
appropriately redefined.
W
o
is defined by (2.4.1) with Y
i
The results of Sections 2.5 - 2.7 also apply.
If the design is indeed balanced, so that V.
-1
then
= o.1
V, 1
-
~
i
~
s,
44
s
max[tr(B )] = max[E _ tr(V ) x C.(w)]
i- l
-w
-i
1
w
w
= tr(V) x max[E~=l 0i Ci(w)].
w
s
Since Ei=l 0i Ci(w) is exactly the term maximized in Section 2.4 to find
the type D region, the type F region is identical to the type D region
if the design is balanced.
It is of interest to attempt to find other combination procedures which are related to the type D and type F tests.
In order to
investigate this issue and to further compare the type D and type F
tests, it is useful to consider the geometrical interpretations of the
characteristic roots,
~1(w), ••• ,~
p
(w), of B.
-w
Restrict consideration
to the two dimensional case (p=2) and let ~1 and ~2 be the two (p X1)
characteristic vectors associated with roots
~1 (w)
and
~2(w).
Assume
that ~l(w) > ~2(w) and that ~1 and ~2 are normalized so that
V'V
-i-i
= 1,
i
= 1,2.
The vectors form a basis of a two-dimensional
Since B
_w is positive definite symmetric, that basis
Euclidean space.
is orthogonal (Graybill [1969, p. 47]).
nates in terms of the usual axes.
Let A be a vector with coordi-
Now consider the ellipse, mentioned
in Section 1.3.1,
A'BA=E,
,. ,. _w-
(3.2.1)
defined by the approximate intersection of the power surface and a
cross-sectional horizontal plane at a height of a + E.
First let A be a
vector along the ':l-axis, so that A = q1 ~l' where q1 is defined so that
(3.2.1) holds.
Then
45
since ~w ~l
q2
= ~l(w)
= '£7~2(w).
~l' whence ql
= '£7~1(w).
Similarly, if ~
= q2
~2'
By definition'~l(w)·~ ~2(w), and therefore ql ~ q2'
implying that the larger the characteristic root, the shorter the axis
of the ellipse in the direction of the associated characteristic vector.
This relationship is pictorially displayed by superimposing the ellipse
(3.2.1) on the
~1~2-axes,
as shown in Figure 3.2.1.
V
~2
Fig. 3.2.1. The Ellipse Formed by the Approximate Intersection
of the Power Surface and a Horizontal Plane at a Height of a + £.
Notice that the type D critical region is given by
max[det(B )]
w
-w
= max(
p
IT
w i=l
~.(w»
1
since the determinant of a matrix is equal to the product of its
p
characteristic roots.
By maximizing
p
IT
iel
~i(w),
one minimizes
IT '£7~i(w) for fixed £, and, therefore, the type D critical region
i=l
provides the test which minimizes the product of the lengths of the
axes of the ellipse A'B A.
--~
On the other hand, the type F region
maximizes tr(B
~i(w).
_w ) = L~-l
1-
For fixed p, the type D critical region
is found by maximizing the geometric mean of the characteristic roots,
46
and the type F critical region is found by maximizing the arithmetic
mean of the characteristic roots.
In the literature there is one other combination of the characteristic roots which is in common usage, one which would provide a
minimax test by minimizing the longest axis of the ellipse A'B A.
--~
This test would maximize the curvature of the power surface along the
axis where the curvature is the least by maximizing the minimum root.
However, for this test it is necessary to find
max
w
(Roy
~
{- t~'(L~=l-~imin
X'X
A
x
Ci(w»~ J.l
(3.2.2)
al. [1970, p. 26]), which is not in general possible in the
case of an unbalanced design.
If the design is balanced, so that
!i = Yi~' 1 ~ i ~ s, then this minimax test coincides with the type D
test.
This can be shown by rewriting (3.?2) as
s
max (Li=l Ci(w) y.) x min
1
A
w
-[
A'V A]
_ - A'A
--
•
(3.2.3)
The second term in (3.2.3) is independent of w, and the solution to the
first term is exactly the type D critical region which was found in
Section 2.4 for the balanced case.
3.3. Bahadur Efficiency
In this section it is shown that, for local alternatives to the
null hypothesis, if there are a large number of finite-sample betadistributed statistics to be combined, the type D critical region provides a test which is superior to Fisher's test according to Bahadur
relative efficiency.
It is also shown that the type D critical region
47
provides a test which is superior to Fisher's test according to a
measure of average Bahadur asymptotic relative efficiency.
Littell and
Folks [1971,1973] have shown that Fisher's test is in general at least
as efficient as every other test when there are a small number of largesample statistics to be combined.
However, other tests have been shown
to be superior to Fisher's test for certain instances in which the two
tests cannot be distinguished on the basis of Bahadur efficiency.
Define the testing problem as H:
A
o
For s
= 1,2, ••• ,
= -0
A versus HI:
A
~
A •
-0
let T be a real valued statistic which depends on the
s
first s observations, {q1, ••• ,qs},of an infinite sequence.
large values of T lead to rejection of H.
s
a
sequence {T } is a
s
Assume that
Also assume that the
standard sequence, which is defined as follows:
Definition 3.3.1. (Bahadur [1960]).
A sequence {T } is said to be a
s
standard seguence for testing Ho if and only if {T } satisfies the
s
following conditions:
(i) there exists a continuous distribution function F such
that
lim Pr o (T s < z) = F(z)
for every z;
s~
(ii) there exists a constant a, 0 < a <
-2 log [1 - F(z)]
00,
such that
= az 2 [1 + 0(1)]
(iii) there exists a function b(A), 0 < b(A) <
for each A
-
~
00
,
as
z~;
such that
A,
_0
l 2
lim pr A(IT s /s / - b(~)
s-+oo
_
If {T } satisfies these conditions, then
s
I
> z) = 0
for every z>O.
48
A= A ,
-0
A '/: A ,
-0
is said to be the approximate slope of {T}.
s
To compare the asymptotic
efficiencies of two standard sequences {T (I)} and {T (2)} with slopes
s
s
cI(~) and c2(~)' respectively, Bahadur [1960] suggested ~12 =
cI(~)/c2(~)
~12(~)
if
as a measure of asymptotic relative efficiency.
> 1, the first sequence is more successful than the second;
~12(~) <
~12(~)
If
1, the second is more successful than the first; and if
= 1, the two sequences are equally successful according to this
measure of efficiency.
To compare the asymptotic relative efficiency of the test based
on the type D critical region to Fisher's test for the testing problem
which has been discussed in this dissertation, the normalized statistics
are used.
If UI, ••• ,U
are s beta-distributed statistics to be com-
s
bined, the locally most powerful test was found in Chapter II to be
S
based on the statistic E = l[y.(a.+b)U./b]
where a.,
b, and Yi are
1
1
1
1
i
defined in Section 2.3.
Let X.1 = (yi(a.+b)U./b)
- E0
(y.(a.+b)U./b)
11
111
(yi(a.+b)Ui/b) - Y so that E (X.) = O.
1
0
1
i
S
are independent, Var o (E.1=
1 X.) = E
S
1
i=
Since the U., and hence the X.,
1
1 Var 0 (X.).
1
1
Therefore, the normal-
ized overall test statistic for the locally most powerful test, based on
the type D critical region, can be written
T (1)
s
s
}
s
1/2
= {E.1= I[X i - E0 (X.)]
I[E.1=
1 Var 0 (X.)]
1
1
c
s
s
1/2
[Ei=l Xi]/[Ei=1 Varo(Xi )]
•
1
Now let Y = -2 log f
i
U
i
eOz
dz, where
eOz
is defined in (1.1.3).
(3.3.1)
Then
49
s
Ei=l Yi is Fisher's overall test statistic for this problem.
Define
(3.3.2)
The asymptotic relative efficiency of {T (I)} to {T (2)} is the
s
s
asymptotic relative efficiency of the locally most powerful test to
Fisher's test, as long as the conditions of Definition 3.3.1 are
satisfied.
The first condition can be satisfied since both T (1) and
s
T (2) are asymptotically normal, so that F =
s
standard normal distribution function.
~
in (i), where
s-.oo
~
00
o
_
(Fisz [1963, pp. 206-207]).
s-.oo
E~=l Yi~ X~s'
is the
The Central Limit Theorem holds
with respect to Ts(l) since the X1.. are bounded and lim Var
s
2
lim Ei=l YI ai/[b(ai+b+l)]
~
(E~1.= 1
X.)
1.
Under H ,
o
and the asymptotic normality of T (2) is well established
s
(Johnson and Kotz, Vol. 1, [1970, p. 176]).
satisfied when F =
~
The second condition is
with a = 1 in (ii) (Bahadur [1960]).
It remains
only to show that (iii) holds for T (1) and T (2).
s
s
Consider first the sequence {T (I)}.
s
s
E =l EA(Xi)/s in probability as s
i
~
00.
The Markov Law of Large Numbers
(Fisz [1963, pp. 216-217]) assures this since
s
2
2
L =l EA(Xi)/s ~ 0 as s ~
i
o.
s
Under HI' E.1.= 1 x./s
~
1.
EA(X~) = 0(1)
Furthermore, since 8.
1.
=
so that
y.e, it can be
1.
50
(3.3.3)
Also,
2
a.b
2 a.+b
1
1
= Li=l Yi (-b-) ----:2=----(ai+b) (ai+b+l)
s
s
2
= L I y. a./[b(a.+b+l)].
i= 1 1
1
(3.3.4)
In light of (3.3.3) and (3.3.4) it can be seen that, for local alternatives to the null hypothesis, so that terms of order 82 and higher
order are ignored,
= s
Notice that 0 < b
l
(~)
-1/2
s
8/2[Var o (L i =1 Xi)]
112
-
(3.3.5)
•
for Xi as defined in this problem.
Next consider the sequence {T (2)}.
s
Under HI'
L~=l
Y/s
~
s
Li=l EA(Yi)/s, again by the Markov Law of Large Numbers since, for this
-
2
problem, EA(Y ) = 0(1).
i
In light of (1.1.2), the density of a non-
central beta-distributed random variable U with parameters band a and
i
i
51
using the exponential expansion.
Therefore,
1
1
s
= Li=l f (-2 log f
o
u
i
(3.3.6)
Substitute (3.3.6) into (3.3.2) and take the expectation to find, for
local alternatives, that
b (A) ~ s-1/2[8/2 L~ 1 E (Y. X.)] x [Var (L~ 1 y.)]-1/2.
2 1.=
0
1.
1.
0
1.=
1.
Eo (Y i Xi) > 0 for all i since E0 (Y i X.)
E0(X.)
- E0(Y.)
1.
1.
1.
-
...,
-
since as Ui increases both Xi and Y increase.
i
-
(3.3.7)
= E0(Y.
X.)
1.1.
Therefore,
-
b2(~)
and
> O.
By combining (3.3.5) and (3.3.7), it is seen that the asymptotic
relative efficiency of the locally most powerful test to Fisher's test,
for local alternatives, is
2
E (Y. X.)] ,
01.1.
(3.3.8)
as long as the limits taken exist.
(In the context of experimental
designs, there are general models where the limits taken in (3.3.8), in
light of (3.3.5) and (3.3.7), exist.
See Sen [1969] for details.)
Also
52
s
s
Eo(Ei=l Y.1 E.1= 1 X.1 )
s
s
s
Eo(Ei=l Yi Xi + Ei=l Ej=l Yi Xj )
i:!j
s
= E. 1 Eo (Y.1 Xi)
ES
E (X.)
_ + E~
1=
1=1 j=l Eo (Y.)
1
~ J
i:fj
since Y. and X. are independent if i :f j and E (X.) = 0 for all j.
1
~
J
J
Combine (3.3.8) and (3.3.9) to see that
(3.3.10)
There is equality whenever Xi and Xi are linearly related.
Since Xi is
a linear function of U. and Y. is a function of U. which is never
1
1
1
linear, the inequality is strict.
There~ore,
(3.3.11)
This result implies that if there are a large number of finite-sample
statistics to be combined, the test based on the type D critical region
is more successful than Fisher's test according to asymptotic relative
efficiency, for local alternatives toR •
o
It should be noted that the asymptotic relative efficiency
based on approximate slopes can be misleading.
It has been noted (for
example, by Bahadur [1967]) that tests which are identical may not have
identical approximate slopes so that
does not occur if
¢12(~)
approximate slopes.
¢12(~)
is not 1.
This dilemma
is defined as the ratio of exact rather than
Assume that the exact distribution function of T ,
s
.
*
F , is such that (ii) in Definition 3.3.1 can be replaced by (ii) :
s
53
(ii)
*
(1) lim F (z) = F(z) and (2) there exists a function f
s
s-+=
on (0,00) into (0,00) such that, for any given sequence
{v s } of positive constants
where
V
° < z < 00,
2/s log[l - F (v)]
s
Then c * (A)
*
*
c * (~)
2
such that lim {v /s}
s
s-+=
-f(z) [1
s
f[b(A)2] is the exact slope of {T }.
-
s
*
cl(~)/c2(~)' then <l>12(~)
ever,
s
=1
+
= z,
0(1.)].
*
If <1>12 (~) =
if the tests are indeed identical.
How-
is often difficult to find, as it is for the sequence
{T (I)}, and hence the use of approximate measures here.
s
Bahadur [1967]
suggests, though, that for local alternatives to the null hypothesis,
<l>12(~)
is likely to be a good approximation of
*
<l>12(~).
Since the
results above are based on the assumption that the alternatives are
local and considering the inherent properties of the test based on the
type D region, it is reasonable to assume that the conclusion of the
superior efficiency of this test over Fisher's test, for local alternatives, is not an artifact.
If the design is not balanced, the final steps in (3.3.3) and
(3.3.6), and hence the measure of relative efficiency given above, do
not apply.
However, similar results can be found for comparisons of
the type F test and Fisher's test.
Let Y
i
=
tr(~i).
Then the approxi-
mate slope of {T (I)}, for local alternatives, is
s
(3.3.12)
This result is analogous to (3.3.5) and follows immediately from (3.3.3)
54
and (3.3.• 4); the 8
1
have not been factored out since the assumption that
0i = Yi 8 no longer holds.
Similarly, in light of (3.3.7), the approxi-
mate slope for {T (2)} can be Written as
s
(3.3.13)
For the unbalanced case, the Bahadur asymptotic relative efficiency,
for local alternatives, is
bi(~)/b;(~)
in (3.3.12) and (3.3.13) respectively.
where
bl(~)
b2(~)
are given
Unfortunately, this ratio is
difficult to evaluate for the unbalanced case.
ratio of the squares of the
and
Consider instead the
average approximate slopes, where the
average is found by integrating over the sphere A'A
= r 2.
The defini-
tion
(3.3. J.4)
is introduced as a measure of average Bahadur asymptotic relative
efficiency, where
f
b. (A)dA.
A'A=r 2 ~ -
Notice that this averaging is the same as that given in Uefinition
1.3.2.1.
f
To evaluate bl(~) and b2(~)' first apply (1.3.2.7) to find
A'A=r2
6 .dA = f
A'V AdA = oCr) x tr(V".) = oCr) x y.
i
A'A=r2 - _i_
-~
~
where oCr) is a function of r.
(3.3.15)
Apply this integration to the right-hand
side of (3.3.12) and recall (3.3.4) to find that, for local alternatives,
55
= s
-1/2
s
(o(r)/2)[Varo(~i=1
Xi)]
1/2
(3.3.16)
•
Similarly, from (3.3.13),
b2(~) = s-1/2(O(r)/2)~s1. __l E (Y. X.)
-
0
1
1
x
[Var
0
(~~ 1 y.)]-1/2
1=
(3.3.17)
1
for local alternatives.
As before, in the present context
o<
It follows from (3.3.14), (3.3.16)-(3.3.17)
b.(A)
<
1 _
00,
i
=
1,2.
and (3.3.8)-(3.3.11) that locally,
Therefore, according to the above measure of average Bahadur asymptotic
relative efficiency, the type F test is superior to Fisher's test for
local alternatives.
As before, Bahadur asymptotic relative efficiency is not
necessarily an accurate comparison of two tests.
Furthermore, the
processes of averaging used above may introduce other inaccuracies or
difficulties in interpretation.
Nevertheless, in light of the method
of derivation of the type F critical region, the above result is of
interest.
3.4. Extensions to Other Problems
Locally optimal combination procedures for tests of hypotheses
specifying the value of a vector of parameters have been found in this
and the previous chapter.
It is possible to find analogous procedures
for combining certain tests of a null hypothesis of the form
H •
o·
= 0,
-
(3.3.4.1)
56
even though the optimality of such procedures is not necessarily
extended from previous results.
The likelihood ratio statistic, A, for tests of (3.3.4.1) is
often written U
, where q,r, and t are appropriate parameters.
q,r,t
Let u(i)
, 1 < i < s, be s independent statistics for testing a
q,r,t
i
common hypothesis; q and r need not be indexed.
Gupta [1971] showed
that in the noncentra1 linear case (that is, when the alternative has
q
unit rank), U(i)
is distributed as IT X.. where X.. has a central
q,r,t
1.J
j=l 1.J
i
beta distribution 130
x
«r+1-j) /2, t/2) , 2
ij
~
j
~
q, and (1 - Xi!) has
a noncentra1 beta density, so that XiI has density
t
00
e
-8 /2
i
=e
where
e.1.
L
h=O
-8 i /2
e.1
il
)r/2+h-1/[2 hhl e(t /2,r/2+h)]
i
00
L (8./2)h 130
Xi!
1.
(t./2,r/2+h)/hl,
1.
1. l
(See Gupta for details of
For the present discussion it is sufficient to note
is dependent on
o
e
(1_x
is the noncentrality parameter.
Under H , X'
g
/2-1
h=O
calculating 8 •
i
that
8~ Xii
*
~'V.~
-
_1_
has density 130
X.
1.1
for some appropriate design matrix V*•• )
-1.
(r/2,t./2),1 < i < s.
1.
--
Therefore, if
(.)
q
(r;t./2) denotes the noncentral density of U. = U 1. t = IT X.. ,
ui - 1.
1.
q,r,
j=l 1.J
where :' = [r/2,(r+I)/2, ••• ,(r+l-q)/2], then
g
e
(r;t /2)= e
ui -
i
-8 i /2
00
8 h I
i
L (-2) -hI g
h=O
.
0
ui
(r/2+h,(r+l)/2, ••• ,(r+l-q)/2;t./2),
1.
o -<
u. < l.
1.-
Recall that the type D and type F test statistics are of the form
(3.3.4.2)
57
where Y is a weight dependent on the design matrix V. and So (b,a.)
i
_1
u
1
i
is the density of a beta-variate U
i
wi~h
parameters band a •
i
Analogously define the following as a test statistic for combining
independent statistics U for testing (3.3.4.1):
i
g
Yi
g
o
U
(r/2+l,(r+1)/2, ••• ,(r+1-q)/2;t./2)
1
i
{3.3.4.3)
0
(r/2,(r+l)/2, ••• ,(r+l-q)/2;t /2)
u.
i
1
*
where Y is a weight dependent on the design matrix _1
V.• If the funci
o
0
tions gi = g (r;t./2) are known, then the statistic proposed in
ui -
1
(3.3.4.3) can be calculated.
known.
For certain cases, the functions g~ are
1
See, for example, Gupta [1971] for the density of U
when
q,r,t
q = 2,3,4, or 5.
Notice that if q = 1 or r =1 , ~_qXr reduces to a
vector and the likelihood statistic is distributed as a single betadistributed variate which is noncentra1 under the alternative hypothesis.
In this case the results derived earlier apply directly.
Assume again that the null hypothesis specifies the value of a
vector rather than a matrix.
Results similar to those in the previous
sections and chapter can be derived if it is assumed that the statistics
to be combined, X , ••• ,X ' have chi-squared distributions which are
1
s
noncentra1 under the alternative hypothesis.
chi-squared distribution with 2b
i
If X. has a noncentra1
1
degrees of freedom and noncentra1ity
parameter 6 , then the density of Xi is given by
i
-6 /2 -x/2 b i
00
bi+j-l j
2j
[e i
e
/2] ~j=O xi
6i /[r(b i +j)2 j!],
x > 0,
where r(a) = (a-I)!. It is easily shown that the type D and type F
statistics for combining
~,
••• ,Xs are of the form
58
(3.3.4.3)
where Y is a weight dependent on the design matrix _1
V. as before and
i
x2 (2b i )
denotes the density of a central chi-squared variate with 2b
i
degrees of freedom.
Rowever, the chi-squared-distributed statistics which arise in
real problems are often statistics used in the testing of (3.3.4.1).
In particular, Sugiura and Fujikoshi [1969] have shown that the nonnull distribution of -2 log A, where A is the likelihood ratio statistic
mentioned above, can be approximated by a linear combination of non-2
central chi-squared-distributed variates, up to order N where N is the
sample size.
For large samples, the distribution is approximately a
chi-squared-distribution which is noncentral under the alternative.
In that case the statistic (3.3.4.3) can be used to provide a combined
test, with Xi
= -2
log Ai' 1
is more complicated.
~
i
~
s.
For smaller samples the result
The overall statistic is similar to (3.3.4.2),
with the central densities there replaced by the appropriate linear
combinations of central chi-squared densities.
Similarly, Siotani
[1971] has given an asymptotic expansion of the nonnull distribution
2
-2
of the Rotelling-Lawley T up to order N
o
•
Again the expansion is a
linear combination of noncentra1 chi-squared variates which can be
approximated by a single noncentra1 chi-squared variate for large
samples.
The extensions here follow in the same fashion as those
involving combinations of likelihood ratio statistics.
Finally, Pillai and Young [1971] have considered approximations
2
to the density of T for the central case. They have proposed that the
o
2
central density of T be approximated by a generalized type II beta
o
59
variate having density
O~u~l.
If research extending this result shows that the noncentral density of
2
T could be approximated by a noncentral beta-type distribution, a
o
procedure for combining s such statistics could easily be determined
along the lines of the above extensions.
CHAPTER IV
APPROXIMATE CRITICAL VALUES FOR TYPE D AND TYPE F TESTS
In Chapters II and III the type D and type F critical regions
for tests of hypotheses specifying the value of a vector were developed.
Formulas which can be used to tabulate critical values for two-sample
problems were found in Section 2.5, and some tables of critical values
are presented in Chapter V.
Although it is, perhaps, reasonable to seek
exact solutions for three-sample and four-sample problems, as the number
of samples to be combined increases, both the number of cases to be
considered in performing the integration and the complexity of the
integrated results increase.
Furthermore, the computer costs of
creating the tables of critical values increase rapidly as the calculations become longer.
It is possible, however, to use approximate rather than exact
critical values in tests combining more than two samples.
The type D
s
and type F test statistics are both of the form Li=l fiU i , where U is
i
beta-distributed with parameters band a., and f
~
an appropriately defined constant.
s
are bounded and Li=l Var(fiU i )
~
00
applies (Fisz [1963, pp. 206-207]).
Therefore, for all real z
i
= (a.+b)Y./b with y.
~
~
Since the s statistics U , 1
i
~
2i 2
with s, the Central Limit Theorem
s,
61
Pr(Z < z) = Pr
.
-
[
~~=l
[fiUi - EO(fiU i )]
.
,
<
s
l/Z [Ei=lVar (fiU i )]
o
]
Z
-..;>
~(z)
s-+oo
-
(4.1)
'
s
l/Z
where Z = ~i l[f.U. - E (f.U.)]/[E. lVar (f.U.)]
and where ~ is
=
1. 1.
2 1. 1.
1.=
2 1. 1.
S
the cumulative distribution function of a standard normal random
variable.
It would be useful to determine values of s which are sufficiently large to provide good approximations in (4.1) so that the standard normal tables can be used to provide approximate critical values.
Computer simulations were used to find the minimum
5
necessary to pro-
vide good normal approximations for normalized type D and type F test
statistics for several different testing problems.
Different values of
b, ai' and Y are used to define different combination problems.
i
The algorithm for the simulation is as follows:
1. Define the testing problem by determining b, and the range
2. Set s.
3. Generate s beta-distributed variates.
(The beta variates
were found by transforming two central chi-squared variates,
say Xl and XZ' with Zb and Za degrees of freedom respectively.
When Xl and X are so distributed,
z
B = XJ/(X l + XZ) has a
central beta distribution with parameters band a.
The
variates Xl and Xz were found by summing the squares of,
respectively, 2b and Za numbers which were generated from a
standard normal population.
These normally distributed
numbers were generated by the PL-l version of program VARGEN
62
as installed on the University of North Carolina Computation
Center's IBM 360/75.)
4. Normalize the sum of the s beta-distributed variates in
order to calculate Z in (4.1).
5. Repeat steps 3 and 4 until a total of 5000 values, say Z.,
J
1
~
j ~ 5000, are generated.
6. Perform the Ko1mogorov-Smirnov goodness of fit test at the
.05 level of significance to measure the maximum distance D
between the empirical distribution function of the {Z.}
~
and the standard normal cumulative distribution function
and to test whether or not the values {Z.} are consistent
~
with a standard normal population. (The PL-1 Scientific
Subroutine Package NDTR as installed at the U.N.C. Computation Center was used to determine values of the standard
normal cumulative distribution function.)
If
D ~ 1.36//5000 = 0.0193, the hypothesis that the {Zi} were
likely to have been drawn from a standard normal population
is not rejected.
7. If D > 0.0193, increase s and go to step 3; otherwise, go
to step 1 to define a new problem or else stop.
In Table 4.1 the maximum distance between the empirical
distribution function of the normalized variates Zi' 1
~
i
~
5000, and
the standard normal cumulative distribution function is given for a
number of testing problems.
Y
1
a
•••
= Ys.
It is assumed that a
If Y = Y, 1 ~ i ~
i
1
=
aM
s, Zi is not a function of the Yi's
and hence there is no loss in assuming that Y
i
= Y = 1,
1
~
i
~
s.
63
TABLE 4.1
VALUES OF THE KOLMOGOROV-SMIRNOV STATISTIC FOR CERTAIN
SIMULATION PROBLEMS, ASSUMING CERTAIN PARAMETERS
ARE CONSTANT FOR ALL SAMPLES
b
y
a
s
D
1
1
1
1
1
1
3
4
0.0262
0.0154
1
1
1
1
2
2
4
5
0.0299
0.0152
1
1
1
1
1
1
8
8
8
7
12
20
0.0348
0.0207
0.0182
2
1
2
3
0.0178
2
2
1
1
8
8
7
12
0.0278
0.0145
3
1
8
8
0.0158
3
3
1
1
15
15
12"
16
0.0217
0.0136
Notice that for fixed b, as a increases the s needed to attain values of
D less than 0.0193 also increases.
This is not surprising since the
beta distribution is symmetric if a = b (uniform if a = b = 1, be11shaped if a
=b
> 1) but is skewed if a
~
b.
The parameter a is more
likely to be larger than b in practical applications, and so the table
has been prepared in that fashion.
1
~
i
able.
~
However, if
ai = a
and Yi = Y,
s, then the parameters a and b are algebraically interchange-
For example, from the table it can be concluded that if b
=
2
and a = a
= 1, 1 ~ i ~ s, s = 5 is the minimum s which is large enough
i
to provide a satisfactory approximation to normality, according to the
criteria previously established.
It can also be concluded, considering
64
the relative skewness of the beta density functions involved, that
since s must exceed 12 if b
=3
and a = 15 before approximate normality
is achieved, s must be quite large indeed if a = 15 and b < 3.
Table 4.2 indicates results of simulations when it is not
assumed that the Y and the a are necessarily constant.
i
i
r = (Y1 '···'Ys )
and
=
~
(a 1 , ••• ,a s ).
Let
Other simulations indicated, as
expected, that the approximate normality was very. sensitive to the
inclusion of large a. when b was small.
1
While this table is i11ustra-
tive of some effects of having varying Yi and ai' the exact parameters
of the Y and a would have to be known before a reasonable guess at
i
i
the minimum s which is large enough to produce good approximations to
normality could be made.
TABLE 4.2
VALUES OF THE KOLMOGOROV-SMIRNOV STATISTIC FOR CERTAIN
SIMULATION PROBLEMS WITHOUT RESTRICTION ON THE
CONSTANCY OF PARAMETERS
b
...Y
a
s
1
12
0.0137
D
2
(1.0,1.2,1.4,1.5,
1.6,1.8,2.0,2.2,
2.4,2.5,2.6,3.0)
2
(1.0,1.5,1.75,2.0,
2.25,2.5,2.75,3.25)
(1,4,5,7
8,10,11,14)
8
0.0207
2
(1.0,1.2,1.4,1.5,
1.6,1.8,2.0,2.2,
2.4,2.6,2.6,2.8)
(1,2,3,4,
5,6,7,8,
9,10,11,12)
12
0.0189
2
..1
(1,2,5,7,
8,10,13,15)
8
0.0145
·I
CHAPTER V
TABLES OF CRITICAL VALUES FOR TYPE D AND TYPE F TESTS
AND A NUMERICAL ILLUSTRATION
5.1. Tables of Critical Values for
Type D and Type F Tests
Tables of critical values for type F and type D statistics have
been prepared for certain parameter values.
Assume that U , i
i
= 1,2,
has a central beta distrihution with parameters band a. and let
1
f
=
(a l +b)y l /[(a 2+b)y 2 ] where Y has an appropriate definition (see
i
Sections 2.4 and 3.2).
for a
f
=1
Table 5.1.1 gives values ofk which satisfy
= .01, .05; b = 1,2; a.1 = 1 (1) 10 (2) 30, i = 1,2; and
(.5) 2.5.
Incomplete tables are also given for b
= 1.5.
By
comparing, for example (2.5.3.2) and (2.5.3.3) for Case 3, it is clear
that there are generally more terms in the series to evaluate if b is
not an integer but a
chosen.
l
and a
2
are integers, for the range of parameters
Since there are more terms to evaluate, the tables for b
are substantially more expensive to compute.
= 1.5
By the time the Case 3
runs for b = 1.5 had been completed it was evident that sufficient
computer funds to complete the table were not available.
Solutions for values of k given aI' a , b, and f were found
2
iteratively for the specified value of a, except for the few instances
in which direct. solutions exist.
Iteration started with an initial
TABLE 5.1.1
CRITICAL VALUES FOR TWO-SAMPLE TYPE D AND TYPE F TESTS
b = 1.0 ex = 0.01
al
f
1
2
3
'I
,
6
7
8
.,
a2
10
12
III
1 ..
18
20
22
26
28
50
1.37'1 1.316 1.271
1.~31 1.77~ 1.733
2.29~ 2.2"5 2.205
2.771 2.720 2.682
2 1;0
1.5
2.0
2.5
1.68' 1.505 1.369
2.093 1.894 1.758
2.~C.7 '.300 2.1f7
2.'2r, 2.717 2.590
1.271
1.666
2.oe2
2.511
1.200
1.602
2.025
2.460
1.1"7
1.556
1.966
2.426
3 1.0
1.5
2.0
2.5
1.553
1.894
2.248
2.611
1.369
1.695
2.044
2.407
1.139
1.476
1.840
2.217
1.070
1.417
1.788
2.169
1.019 .982 .'53 .932 .915 .890
.80;0 .1143 .A37 .A~2 .A28
.872 .1160
.8;>5 .822
1.376 1.346 1.'23 1.306 1.292 1.271 1.257 1.'46 1.2'A 1.'31 1.226 1.~22 1.210 1.21~ 1.21'
1.75' 1.726 1.706 1.690 1.678 1.659 1.646 1.~36 1.6'8 1.622 1.617 1.';'13 1.609 1."~6 1.6C4
~.136 2.111 2.092 2.078 2.06t; 2.048 2.036 ;>.026 2.019 2.013 ;>.oor. 2.004 2.001 1.99P 1.~96
II t.O
1.5
2.0
2.5
1.451
1.740
2.0 4 4
2.357
t.271 1.1~9 1.044 .974
.92:'1 .1185 .~56 .834 .1'117 .791
.773 .760 .7'50 .7 4 3 .n7 .7'12 .721$ .724
.721
1.545 1.~15 1.328 1.269 1.226 1.197 1.174 1.1~7 1.142 1.121 1.106 1.096 1.0n7 1.C91 1.075 1.071 1.067 1.0~4 1.~t;1
1.6 4 3 1.7~0 1.6~2 1.569 1.553 1.~26 1.506 1.49C 1.477 1.'158 1.44~ 1.~34 1.~27 1.420 1.415 1.411 1.408 1.40"' 1.~02
2.156 2.039 1.967 1.91 9 1.86~ 1.860 1.842 1.827 1.815 1.797 1.784 1.775 1.7~7 1.761 1.756 1.752 1.749 1.74~ 1.744
5 1.0
t.5
2.0
1.374 1.200 1.070 .974
.903 .851 .811 .7P.l
.7~8
.739 .712 .694
.61\0
.670 .662 .~56 .6~1 .1'046
.643
.640
1.623 1.431 1.302 1.214 1.154 1.111 1.079 1.055 1.0~7 1.022 1.000
.985 .974
.91'.5 .959 .953
.9 4 8 .94r:,
.941
.93'1
4
1.~85 1.687 1.5~3 1.4A4 1.431 1.393 1.365 1.'144 1.3~A 1.315 1.~~5 1.282 1.271 1.~h3 1.257 1.~52 1.24~ 1.2 4 1.241 1.~~q
2.157 1.Q58 1.841 1.767 1.718 1.683 1.658 1.639 1.623 1.611 1.r,93 1.580 1.~70 1.~h3 1.557 1.552 1.548 1.~44 ~.'5u, 1.~39
:o.~
2.'
2.~
1.2'5
1.5f3
1.918
2.2 11 9
1.107
1.523
1.958
2."01
1.'35 1.206 1.182 1.1'" 1.119
1.~"
1.673 1.651 1.618 1.595
2.173 2.149 2.12~ 2.096 2.077
2.~5~ 2.6~9 2.611 '.'83 2.563
1.077
1.4"
1.938
'.382
1.099 1.0~" 1.072 1.062
1.~77 1.'~4 1.~54 1.5'"
2.061 2.00;0 2.~40 2.033
2.5"9 2.5~e 2.~30 2."4
211
1.e5., 1.689 1.'~3 1.451
2.327 2.1"" 2.005 1.9011
2.eco 2.609 2.4~6 '.369
3.27~ 3.07~ 2.~~8 2.8"0
1 1.0
1.5
1.05" 1.0"7
1.5~8 1.533
2.027 :0.023
2.519 2.514
1.0"2 1.037
1.~2~ 1.~24
2.019 2.01~
2.511 2.~C~
1.053 1.034 1.007 .9S8 .'67 .'<;6 .9'5~ .953
.948 .941.6
.94~
.'37
1.~~0 1.466 1.445 1.430 1.'119 1.411 1.405 1.399 1.395 1.391 1.3=P 1.~e6
1.9~2 1.~09 1.~90 1.077 1.1I~7 1.1\59 l.e~3 1.p4e 1.844 I.A40 1.e~7 1."3"'
2.~67 2.~55 2.337 2.325 2.~15 2.308 2.'02 2.297 2.293 2.290 2.~P7 2.~~4
6 1.0
1.5
2.0
2.5
1.147
1.344
1.761 1.56'1
2.00n 1.802
7 1.0
1.5
2.0
2.5
1.271
1.460
1."'63
I. Q 74
8 1.0
1.5
'.0
2.!i
1.23'1 1.077 .~~3 .856
.723 .678 .1'.43
.7Pl
.6'6 .~94 .~62 .539
.'\23
.480;
.'512 .~02 .49'1 .490
.4~1
.477
1.4~3 1.223 1.094 1.002
.936 .8011 .8,,1
.1\24
.M3 .786 .761
.744 .731
.1"2 .7:4 .709 .704
.700
.6':'6 .f,'l3
1.583 1.392 1.266 1.181 1.1~2 1.060 1.049 1.0;>6 1.000 .')93 .97;>
.957 .945 .9~7 .,.30 .925 .9?1
.917 .914
.911
1.772 1.~77 1.~55 1.376 1.322 1.284 1.256 1.23~ 1.219 1.206 t.186 1.17;> 1.162 1.154 1.148 1.142 1.138 1.135 1.132 1.129
,
1.205 1.053
1.0
I.";
2.0
2.5
10 1.0
1.~
2.0
2.5
1.31~
1.~32
1.019 .923 .e.Sl
.796 .755 .723 .691\ .679 .650 .630 .1'.16 .606 .5'17 .'591
.586 .'581
.577 .'574
1.215 1.126 1.063 1.019 .986 .9"'1
.941 .9?6 .903 .667 .676 ;1l~7 .~60 .~~l.6 .8~0 .1\46
.843 • B4,':'
1.441 1.360 1.304 1.265 1.237 1.215 1.1"6 1.185 1.165 1.150 1.140 1.B2 1.12(, 1.120 1.116 loll? l.l~9 1.107
l.f83 1.608 1.557 1.522 1.49(, 1.476 1.4;0 1.448 1.429 1.416 1.406 1.3"8 1.'92 1.~07 1.3113 1.380 1.377 1.374
1.107 .9A2
.885 .811
.755 .712 .678 .652 .613 .601
.580
.565 .554 .~45 .538
.533 .528 .~2'+ .521
1.276 1.147 1.057 .9 9 2 .946 .911 .111\"; ./1<,5 .049 .1125 .11011 .7% .7~7 .71\0 .774
.770 .71:a6
.762 .71,;'
1. 4 69 1.344 1.2"'0 1.2n3 1.163 1.133 1.111 1.0q3 1.079 1.059 1.044 1.0:'13 1.025 1.019 1.013 1.0n9 1.005 1.00' .999
1.678 1.557 1.4eo 1.4~9 1.391 1.364 1.:'144 1.3~8 1.315 1.~96 1.282 1.<72 1.264 1.'58 1.253 1.249 1.245 1.242 1.240
.9::'2
.834
1.0~3
.9~9
.758 .69!1 .652 .616 .5P7 .564
.8"11
.752
.641 .r03 .774
.734
1.519 1.33n 1.2C3 1.116 1.n55 1.01~ .,:eo .9;;5 .Cj~f, .nl
1.6rA 1.495 1.371 1.290 1.235 1.196 1.167 1.145 1.128 1.115
1.30;~
1.1~0
.530
.7OR
.lIq"!
1.09~
1.1e2 1.03~ .915 .817 .7~9 .679 .613 .5"4 .5"4 .~40 .504
1.317 1.14(, 1.019 .925 .1\55
.764 .733 .7] 0 .691
.1lO~
.664
1.4"4 1.279 1.151 1.0f3 1.000
.95'; .922 .1197 .877 .861 .838
1.618 1.426 1.302 1.219 1.162 1.122 1.092 1.069 1.052 1.030 1.017
.506
.690
.4 Q 9
.477
.6(,7
.Bln
.P72 .EI';4
1.060 1.070 1.052
.479
.645
.822
1.CO~
.~77
• ~ (,1
.L+-1IB
.~32
.621
.1\10
.992
.'la3
.8~1
.467
.460
.654
.454
.649
.449
.645
.P~l
.1'57
.?1.47
.q u 3
1.0~5 1.050 1.046 1.042
.~~O
.~38
.<,1 4
.794
.'177
.430
.607
.789
.972
.423
.602
.79"
.967
.641
.441
."'3A
.S4r,
.~37
.4lot"
1.0~" 1.~36
.418
.5"18
.a:"
.7~O
.777
.'%1
.96~
.'59~
.41~
.";91
.774
.958
0\
0\
TABLE 5.1.1 (Continued)
b = 1.0 a. = 0.01
al
f
6
7
8
,
.791
.712
.650
1.1"5 1.007 .890
.117'1 .802
1.257 1.09" .970
.7"6
1.37~ 1.201 1.07'1
.982 .916 .868
1.~0~ 1.321 1.1'15 1.109 1.0_9 l.OOE.
.601
.70'1
.832
.97'1
.56?-
.5~0
.1105
.'50
1'1 1.0
1.5
<'.0
2.5
1.119 .'le!\ .e72
.773
.1140
1.213 1.0S A .9~7
1.317 1.1"5 1.019 .926
1."2Q 1.2"5 1.118 1.030
.6"'+
.765
.P57
.966
.630
.707
.80F.
.921
.~co
.662
.767
.ae7
16 1.0
1.5
2.0
2.5
1.t,9Q
18 1.0
1.'1
2.0
1.0e~
1
12 1.0
1.5
2.0
2.5
~.5
2
.967
1.leo 1.032
1.2n 1.H4
1.3~6 1.le9
3
./IF'0
.913
.9110
1.06~
.850
.8'5
.951
."56
1.013
1.233 1,073
1.317 1.144
1.0~9
1.1~5
It
5
._03
.383
.53"
.(,1\9
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.600
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.'+37
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1.1911 1.044
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1.042 .940
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1.12" .994
1,179 1.030
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1.037 .937
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1.117 .'lIlF.
1.103 1.019
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2.5
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.51~
.".11
.237
.312
.231
.3ce
.3'lC
.1f7e,
• ~~,c
.~~4
.437
.527
.277
.3 4 2
.4111
.'iCO
.264
.332
.410
.493
.253
.32"
.41l7
.2 ....
.317
.3911
.4113
.255
.316
• 3P.7
.463
.2 ....
.305
.361
.4sa
.23">
.301
.375
.453
.227
.295
.371
.2""
.303
.237
.29"
.361
." 32
.227
.267
,355
.427
.21'1
.231
.283
.221
.275
.338
... 05
.257
• :'\04
.3(,1
.t42'"
.~61\
,431:1
.2"3
.;"l?2
.351
.4:6
•.~"14
."10
.4~O
,lot ~ ..
.~60
.?96
.3.,\
.3'5~
.... 5
.404
.'!33
.~q4
.,+7Q
.'ic.q
.!21'
.41~
.-.
.C::£ •
.2 Q l
'3~7
." 4
">
.3~a
.213
.276
.3"6
.&423
.&41q
.213
.269
.206
• 2~ ..
.2~1
.33~
.3~Q
.4(10
.397
0\
"
TABLE 5.1.1 (Continued)
b = 1.0 a = 0.05
az
alf
3
,
9
12
1'1
16
1 1.0
1.5
:>.0
2.5
1.68'1 1.'69 1.331 1.2'12 1.182 1.139 1.108 1.085 1.0~7 1.053 1.033
2.113 1.892 1.760 1.678 1.625 1.588 1.562 1.~'13 1.528 1.517 1.502
2.~~3 2.331 2.2~5 2.129 2.082 2.050 2.026 2.012 2.0na 1.191 1.911
3.000 2.119 2.(,~9 2.~90 2.5111 2.519 2.500 2.'106 2.'175 2.'166 2."52
1.019
1.'192
1.961
2.11'12
1.010
1.'18'1
1.959
2.43'1
2 1.0
1.1If'.9 1.260 1.129
1.ec" 1.~911 1.1176
2.1~7 1.953 1.6U1
2.521 2.326 2.226
1
1.~
2.0
2.5
2
5
(,
7
8
10
18
20
22
2-
26
28
30
.9811
.9'12
1.003
.998 .993 .990
.981
1.'178 1.'173 1."68 1.465 1.1I~2 1.II~Q 1.1151
1.953 1.9"'1 1.~1I3 1.9'10 1.937 1.9!~ 1.'152
2.'128 2 ... 23 2.418 2.'115 2.1112 2."09 2."01
1.C'l1 .993 .957
.930
.911
.6"6 .893 .8,,5 .852 ./1'12 .835 .Po29 .92'1 .620 .61" .8U • ell
1.'101 1.~62 1.331 1.306 1.291 1.27 7 1.2~6 1.2119 1.237 1."8 1.221 1.~15 1.211 1.207 1.'011 1.201 1.191'
1.71'11 1.142 1.713 1.692 1.675 1.662 1.~Sl 1.635 1.62'1 1.615 1.~r.8 1.603 1.598 1.5 0 '1 1.<;91 1.51'''1 1.'iA6
2.165 2.126 2.098 2.017 2.061 2.0'18 2.03A 2.022 2.011 2.003 1.9"6 1.'190 1.986 1.982 1.919 1.916 1.97'1
.6110
1.0(10
.79'1
.113
.7'13
.7211
.710 .699
.685
.911
.861
.623
.15 7
.6"2
."1'19
.9911
1.273 1.201 1.154 1.121 1.096 1.076 1.0~3 1.052 1.03'1 1.022 1.0t2 1.0n5
1.560 1.502 1.1159 1.112n 1.1106 1.389 1.375 1.36'1 1.3'17 1.335 1.326 1.319 1.31'1 1.309
1./172 1.810 1.769 1.739 1.718 1.701 1.6n8 1.678 1.662 1.650 1."'11 1.6~'1 1.~2" 1.62 u
.616
.669
.e'iJ7
.672
.9P.14
.987
.9Al
1.302 1.29" 1.297
1.617 1.61~ 1."12
3 1,0
1.5
2.0
2.5
1.331
1.59'1
1.875
2.169
1.129
1.390
1.677
1.976
,
1.0
1.'5
2.0
2.5
1.2112
1.452
1.600
1.921
.51'9
1.0'17 .917 .830 .;71
.699
.676
.6'i6
.6'111
.623
.597
.'iA2
.571 .572
.56'1
.!.6'S
.5!>3
.729
.6011
.1134
.89/\
.1I6l,
.A4'1
.0113
.038
1.251 1.131 1.056 1.006
.926
.'110
.680
.A57
.0.31
.p"o .P2&
.971
.9'15
1.1182 1.370 1.301 1.2~6 1.224 1.201 1.1A3 1.168 1.1~7 1.140 1.128 1.111\ 1.111 1.105 1.101 1.0'17 1.093 1.0~1 1.r.F.1I
1.726 1.620 1.S55 1.512 1.4112 1.460 1.1142 1.4?'1 1.'118 1.'102 1.390 1.~81 1. 3"'1 1.'68 1.364 1.360 1.3S7 1. 3S4 1.!~1
5 1.0
1.5
2.0
2.5
1.11'2
6 1.0
.4115
.1140
1.139 .9!17
.611l
.537 .521
.496
.467 .458
.451
.436
.823 ".129 .66'1
.583 .557
.480
.1130
."'~~
.63'1
.879 .82'+
.l,50
1.2112 1.0116 .9f'.1
.784
.756 .734 .71 7
.703 .683 .669 ."!l8
.6 4 4 .1;311
.b~O
.62 7
t"~~
.829 .82l,
1.441 1.2113 1.125 1.051 1.0Cl
./\52
.844
.833
.966
.940
.921
.906
.893 .875 .862
.1138
.8< 3
.~20
1.611 1.41'1 1.302 1.233 1.187 1.15'1 1.130 1.112 1.098 1.066 1.069 1.056 1.0'17 1.040 1.03'1 1.029 1.025 1.022 1.019 1.017
1.~
2.0
2.5
.4'17
.1I9~
.5/\8
.'\23 .51 4
.u87
.993 .861
.771
.709 .664
.632
.f'.07
.573
.!l50
.53'1
.'i07 .501
.1193
.7'19
.720
1.15'1 1.0~2 .953
.'100
.At5
.729
.724
• 71 ~
.767
.743
.663
.836
.786
.753
.735
.71~
.711
r
.Q~~
.9 ,9
.9~~
.'I3l,
1.'\42 1.3'14 1.229 1.157 1.110 1.076 1.052 1.0~3 1.018 1.006
.989 .976
.9~6
.948
.'153
.941
1.7'12 1.5117 1.437 1.370 1.326 1.295 1.271 1.25'1 1.2'10 1.'-29 1.212 1.200 1.190 1.1113 1.178 1.173 1.169 1.166 1.163 1.161
.'1~8
.930
.699 .632
.SA:'I
.547
1.106
.7"4
.520
.(,95
.~72
1.:'28 1.037
.1125 .766
.725
.910
.A3~
1.364 1.1611 1.0u7
.~7C
.918 .882
.855
1.0;11 1.314 1.200 1.128 1.0nO 1.04l, 1.021 1.00'-
.'181
.63'1
.1106
.'376
.456
.f.18
.71\7
.958
.438
.603
.773
.945
.425
.592
.763
.'135
.415
.5AU
.755
.928
.1107
.'i77
.749
.92?
.401
.60;4
./\19
.9A8
e
1.0
1.5
2.0
2.5
.773
.676
1.085 .911
1.1FII 1.000
.783
.871
.9.8
1.305 1.111
.908
1.'133 1.237 1.120 1.0116
1.0
1.5
2.0
2.5
1.067
10 1.0
1.5
2.0
2.5
1.~0~
1.6~1
1.3~~
7 1.0
1.5
2.0
2.5
,
.~qO
.74'1
.917
.396
.567
.740
.Q!3
.392
.389
.38l,
• ~f4
•
.737
.'11u
.5":
.73u
.%7
.731
.905
.'57
.'\10
.665
.821
.354
.5C7
.662
.81A
.350
~
1&':8
.607
.72?
.85'1
.996
.557
.679
.816
.961
.520
.6'17
.788
.935
.491
.(,2:'1
.767
.'116
.4l,9
.604
.7..,0
.900
.1151
.589
.737
.888
.42'1
.567
.717
.870
.405
• 5.51
.703
.856
.392
.540
.l,'12
.8'17
.381
.531
.6A5·
.639
.373
• ..,2'1
.(,78
.1133
.367
.673
.628
.362
.514
.669
.82'1
.427
.5u9
.6ill
.817
.399
.52(,
.660
.798
.379
.510
.6u6
.78'1
.365
.498
.635
.77'1
.35'1
.IIA9
.339
.47l,
.f15
.756
.33'1
.471
.611
.751
.329
.u67
... 07
.74n
.325
.11(,4
.60U
.7U5
.322
.1161
.~'7
.7~7
.346
.u81
... 20
.761
.407
.51fo
.635
.75A
.379
.358
.475
.599
.725
.343
.463
.33'.1153
.579
.7n7
.323
.446
• ..,73
.700
.316
.310
.1135
.5fo3
.691
.306
.u31
.55<;1
.f'.Be
.302
.'1211
.298
.420,
.553
.6112
.757
.tlul
.972
1.259 1.eM. .942
1.371 1.176 1.056
.658
.751
.A60
.980
.588
.foS8
.803
.929
.537
.l,43
.76'1
.892
.498
.610
.734
.866
.469
• 'ill !I
.712
.845
.11115
.1"11:'1
1.053
.7113
1.131
.951
.818
1.221 1.031
.906
1.321 1,127 1.006
.6'14
.725
.821
.927
.'\73
.661
.762
.874
.521
.614
.721
.1I3l,
.401
.580
.691
.809
.'151
.427
.533
.896
1.1~6
.~72
.S5~
.f'.l,7
.7/\/\
.5~5
.6 5
.830
0
• 6,~q
.771
."92
.(,14
.739
.~Ae
.715
.519
.440
.~67
.6'35
.~C)~
.~e~
.~a"
.660
.!H5
.c,~2
.742
0\
00
TABLE 5.1.1 (Continued)
b = 1.0 a = 0.05
81
f
82
2
1I
II
5
6
7
a
9
10
12
1"
16
18
20
22
2"
26
28
50
.7211
.783
.852
.9:!'1
.625
.687
.763
.848
.550
.620
.701
.7':'1
... 96
.57t
.657
.751
."56
.535
.62"
.721
."2"
.579
... 67
.565
.657
.3.. 11
..... 1
.5 .. 2
.647
.326
."23
.526
.632
.510
.la09
.5tla
.1;21
.613
.289
.391
.la98
.f.07
.281
.385
.493
.601
.275
.3AO
.~C?1
.399
."115
.5/10
.6111
.2'11.'1
.3'19
2.~
.865
1.C"3 .'n9
1.166 .'J8t
1.2116 1.055
.270
.376
.484
.593
.266
.372
.4!1
.!.9C
.2(,2
.3f.9
.la7/l
.'5/11'
1_ 1.0
1.5
2.0
2.5
1.01 ..
.852
1.068
.897
1.H7 .9 .. a
1.193 1.006
.710
.5:!,4
... 80
.541
.612
.690
... 38
.503
.577
.659
... 05
.117"
.0;0;1
.f.35
.379
.110;0
.530
.616
.35a
."32
.51"
.601
.326
.404
."89
.579
.30 ..
.385
.472
.56"
.287
.27"
.3(,0
.40;0
.544
.26"
.351
.4 .. 3
.0;37
.256
.2,.4
.335
.1128
.23'1
.331
.437
.532
.2 .. 9
.339
.432
.5:>7
.23'5
.815
.879
.608
.661
.723
.7q3
.4~ ~
.~23
.520
."22
.517
16 1.0
1.5
2.0
2.5
1.010
1.0""
.6 0 9
.7 .. 1
.:'192
.343
."06
.476
.551
.310
.377
.1150
.528
• 287
.356
... 32
.512
.'F-9
.'''5
.321
... 01
.2t?~
.Z::H~
.~-96
.~E;t
.4('q.
.230
.308
.390
... 73
.22 ..
.303
.l4q1
.237
.314
.395
... 78
.215
.341
.1119
.500
.2"6
.3'0
.409
.219
.0;15
.587
.365
.4'5
.4 rt 3
1.1"4
.""2
.1'1\1
.923
.971
... 69
."6f,
.!1q
.41>3
11.'1 1.0
1.5
2.0
2.5
1. 003
1.0"6
1.077
1.12 ..
.'98
.355
."19
."88
.2H
.33 ..
... 00
.1170
.'''6
.318
.3116
.1158
.2"2
.376
.1149
.'31
.,97
.368
.11 .. 1
.;>;>2
.529
.332
.3Ar,
..... 6
.512
.361
... 35
.215
.2A3
.356
.4:!O
.20'3
.27!!
.352
.1126
.2'0'"
.274
.3 .. Q
.1123
.199
.'71
.3115
.1&20
.t?~!'
.323
.:370
."23
."81
.289
.339
.245
.300
.:"(,1
.425
.2 .. 1
.2"8
• 3"0
... 15
.219
.278
.3 4 1
.407
.210
.270
.33 ..
.401
.203
.26"
.329
.396
.197
.259
.1'11
... 56
.26"
.317
.375
... 38
.~Z~
.32C
.38 P
.187
.25C
.317
.281
.325
.375
.429
.256
.302
.355
.411
.237
.2116
.340
.397
.2'2
.'273
.3:>1\
.31'>7
.210
.262
.319
.193
.2u8
.~79
.201
.25 ..
.:!'12
.372
.:!67
.187
.242
.302
.!63
.275
.3111
.359
.408
.2 .. 9
.291
.33/1
• 389
.':'10
.'74
.322
.37 ..
.215
.2(,0
.310
.3(,4
.'03
.:>50
• 301
.355
.193
.241
.29"
.3'19
.1115
.235
.2P8
.3113
.179
.229
.2 8 3
.339
.17~
.16~
.2'"
.279
.33~
.220
.21",
.!32
.22"
.'6"
.3011
.355
.209
.2'10
.2'l'l
.3411
.197
.'39
.335
.187
.230
.278
.329
.179
.223
.21'2
.323
.172
.218
.2(,7
.31&
.166
.213
.2f'
.314
.161
.208
.2':;9
.311
.191
.,30
.181
.221
.173
.21"
.;>59
.306
.16&
.HO
.20~
.203
.24'1
.2'l7
.150;
.199
.293
.155
.1910
.237
.251
.150
.190
.233
.;>711
1
12 1.0
1.5
2.0
1.035
.7~9
.S~2
.659
.733
.507
.600
.~70
.460
... 0;2
.5~5
.523
.571
.467
.8 .. 2
.597
.642
.69"
.752
.~27
.6~0
.579
.6"5
... 25
.479
.543
.612
.835
.1'61.'1
.905
.9116
.6'12
.7'8
.768
.81 11
.509
.627
.672
.722
.51"
.657
.458
.502
.553
.611
.415
... 62
.516
.57&
.:'IAl
.430
... 87
• '1119
1.025
1.060
1.101
.829
./l51l
.8 C1 1
.92S
.6Q5
.717
.507
.5113
.5115
.632
.1151
.4(\'1
.53/'
.583
.407
.448
.495
.548
.373
.416
.7"2
.562
.616
.655
,£.98
.520
.3116
.3"'0
.""2
.1198
.82"
./l51
.879
.911
.680
.708
.7"0
.1'75
.577
.607
.641
.679
.SCl
.533
.570
.S11
.4"5
... '"
..... e
.3:!,9
.378
.510
.56'
.401
... 37
.479
.525
.367
... 04
2.~
.'l93
1.017
1.04!!
1.083
.4~'+
.31(,
.357
."05
.4~6
."7"
."56
211 1.0
1.5
2.0
2.5
.990
1,011
1, 03"
1,%8
.820
.1'76
.7111
.7'1.0
.7"'1
.572
.4'l7
.525
.558
.595
.11"0
.470
.500;
.54"
.396
... 28
.4(.5
.506
.:'Ifo2
.3'15
.1&34
... 77
.3 ....
.3(,8
.:no
."~9
.l.6~4
.3117
.311'1
."36
26 1.0
1.5
2.0
2.5
.987
1.006
.816
.8311
.672
.569
.593
.620
.651
."93
.519
.5"8
.581
... 36
."63
.119"
.530
.392
... 21
."5"
.491
.357
.:'187
.422
.461
.329
.36 0
.3'17
.437
.306
.338
.31'7
.410
.270
.305
.346
.~90
.2 ....
.281
.:32'+
.310
... 90
.'513
.540
.569
.433
.457
.486
.517
.38'3
... 1 ..
.4 .. 4
... 78
.35"
.31\1
.41 ?
.~25
.302
.331
... 0 ..
.266
.2911
.3311
.375
.23':1
.2n
.312
.354
.219
.;>55
.295
.339
.2n ..
.2111
.283
.3;>7
.4B7
.5011
.430
,:'ISO
.;>62
.'91
.325
.362
.235
.;>(,6
.302
.340
.l'l9
.'lIe
.2~1+
.47a
.507
.298
.325
.357
.392
.'15
.~~3
.386
.409
.436
.467
20 1.0
1.5
2.~
2.5
22 1.0
1.'5
2.0
1.~9~
.~U4
.P70
./l9/\
.7~9
.'7~3
.6~6
.5~9
.630
.6611
1.{l~9
.as,
1.056
.8~'1
.7:?1
.749
.'Ht~
.1113
.B3"
.IISS
.879
.6€-9
.S':'l
• 7l'~
.739
.565
.1\11
.830
.61',7
.6 P E-
.e~o
.708
.731
.563
.503
.60e.
.631
28 1.0
1.5
2.0
2.5
1.0C2
1.022
1, ~II7
30 1.0
1.5
2.0
2.5
.982
.99!l
1.0lF.
1.038
.871
.588
.1'.13
.640
.5~5
.6C3
.560
.51q
,uS:?
..... '1
.uF,6
.44A-
.:'170;
."04
.436
.5~7
.3';4
."06
."F-5
.354
.3 n 7
.1123
.322
.34/1
.37 A
.4\1
.366
.39~
.2115
.325
.3~F-
.272
.313
.'~6
.3~4
.2Aq
.273
.;>6~
.3111
.311
.'22
.262
.\1\7
.177
.213
.2"4
.304
.296
.4~a
.5':'7
.3~S
.168
.392
.253
.301
.161
.:>os
.no
.21.17
.290
.242
.226
.2~1.I
.~2P.
.3P~
.1/l1
.23·
.29/\
.177
.231&
.29"
.3~9
.!:~
.24~
0\
\0
TABLE 5.1.1 (Continued)
b = 1.5 a. = 0.01
81
f
1
2
II
:I
S
,
7
8
,
82
10
12
111
16
18
20
22
2'1
26
28
30
1.000
.9'5
.'e8
.'85
1 1.0
1.S
2.0
2.5
2 1.0
1.5
2.0
2.5
3 1.0
1.5
2.0
2.5
II 1.0
1.5
2.0
2.5
'5 1.0
1.5
2.0
2.5
6 1.0
1.5
2.0
2.5
.
.'8'
7 1.0
1.5
2.0
2.5
.9"'1
8 1.0
1.5
2.0
2.5
.982
,
1.0
1.5
2.0
2.5
.'511
10 1.0
1.5
2.0
2.5
.931
.909
.880
.857
.987
.'62
.9'13
.92'
.918
.90'
.'02
.896
.891
.ee6
.982
.'5'1
.':11
.f'9'
.871
.853
.811'
.828
.819
.812
.1\06
.801
.7'6
.'8'
.,....
.90'1
.880
.857
.821
.796
.777
.7(,3
.752
.7'13
.755
.729
.72'1
.719
.'32
.886
.1l50
.8~0
.7'6
.760
.73..
.715
.700
~990
.666
.960
.679
.972
.671
.965
.665
.960
.660
.9StS
.655
.951
.886
.850
.8'10
.803
.1103
.712
.1'1n
.99.2
.710
.961
.683
.939
.663
.922
.6'16
.909
.636
.g99
.62(,
.891
.618
.S8'1
.612
• '179
.601,
.~02
.87'1
.e10
.592
.1132
.51l2
.823
.57"
.e16
.'i67
.811
.501
.806
.'i57
.1102
.16'1
.'114
.733
.948
.106
.669
.1l95
.6'11
.872
.620
.1155
.6~'1
.921
.842
.620
.99"
.772
• 95'1
.133
.'120
.701
.8'1'1
.675
.1l72
.63'1
.1139
.605
.1116
.5S'I
.198
.51,6
.71''5
.'i55
.77'1
.999
.5'14
.766
• "1
.536
.759
.965
.529
.153
.geo
.523
.7 .. "
.975
.518
.1 ....
.971
.796
.953
.746
.910
.1011
.1176
.675
.6 .. 8
.648
.BU.
.606
.792
.992
.576
.7611
.911
.5511
.70;0
.955
.537
.0;23
.7~6
.72!j
.9 .. 3
.<133
.513
.716
.925
.5011
.709
.919
.1197
.703
.913
.1191
.69/\
.901'
."86
.69'1
.90"
....,
0
..
TABLE 5.1.1 (Continued)
b
1.5 a. = 0.01
al
f
6
7
II
9
10
1?
1'1
16
11l
20
22
2_
26
29
30
.1196
.1l21
• .,113
.760
.811(,
.710
.8111
• .,97
.669
.1I0!l
.'16'
.6311
.776
.':1110
.60(,
.7':'1
.':11'1
.'62
.717
.A66
.530
.691
.865
.'06
."72
• All"
.IIA7
.6'17
.1l36
.1173
• toll 6
.1\2(,
.1162
.637
.1\17
1.000
.11'"
."'9
.'l11
• 9Q ...
.11115
."23
.1105
.'189
.1139
.61"
.900
.9p ..
.11"3
."13
.79(,
.92.0
.871
.96':1
.796
.896
.7311
.831\
.':1611
.683
.791
.922
.6111
.75"
.1'.119
.6~5
.576
.6':1A
.8"1
.':192
.'30
.660
.1196
.633
.78"
.':1'+1
.1170
.(,12
.767
.':125
."'il
.5'17
.7'13
.913
... 36
.'185
.743
.903
... 23
.575
.7311
.895
."1"
.567
.727
.811'1
."05
.561
.721
.1183
.39"1
.~93
.723
.8"2
.515~
.~~1
.716
.87"
.712
• P.75
.1170
.588
.720
."'3
.5'\0
... 07
.'\37
.677
.39"
.527
.668
.1112
.383
.519
.661
.805
.375
.512
.(,55
.1100
.31>11
.506
.650
.795
.361
.0;01
.660
.11 ....
.566
.702
.1143
.371
.350
... 72
... 01
.732
.!4!
."61>
.59':'
.727
.336
.461
.-:91
.723
.316
.421'
.C.""6
...,,6
It
1.0
1.'
2.0
.ge7
1'1 1.0·
1.'
2.0
2.5
.9Eo2
16 1,0
1.5
2.0
2.5
.9_3
111 1.0
1.5
.929
.999
1~
2
a2
,
3
1
2.'
.853
.935
.839
.909
.99"
2.0
.777
.861
.961+
.763
• 83"
.922
2.5
20 1.0
1.5
2.0
2.5
.'318
.979
22 1.0
1.5
• 909
.963
2.0
1.000
.'02
.9"9
2.0
2.5
26 1.0
1.5
2.0
.993
.8 a 6
.939
.9~9
2.'S
28 1.0
1.5
.988
2.0
.O'?l
.929
.97'
2.5
30 1.0
1.5
2.0
2.5
.983
.8P6
.922
.9~3
.963
.663
.75'+
.865
.':190
.620
.715
.1130
.'159
.56"
.663
.802
.933
.5'1"
.657
.780
.913
.o;oe
.700
.77..
.866
.971
.6.. 8
.725
.821
.930
.601+
.665
.7115
.1197
.568
.652
.7'15
.871
.537
.625
.732
.8 .. 9
... 87
.583
."95
.817
.451
.552
.669
.794
.423
.!l30
.':'50
.777
.401
.512
.(,'1'1
.763
.38"
.,,99
."2"
.7'13
."1'1
.745
.360
... 80
.607
.738
.436
.524
.6211
.740
.407
.5no
.6C8
.722
.3e4
.367
.4(,11
• .,eo
... 97
.352
.457
.571
.689
.3"0
..... 7
.563
.61'1
.331
.323
.41l2
.'192
.7011
.440
,434
.'i56
.676
,551
.670
.325
.421
.">26
."34
.315
... 13
.'\19
.628
.306
.40.,
.513
.299
."01
.62~
.619
.715
.802
.90'1
.617
.7'+5
.1'82
.(,1'8
.11'10
.1\20
... 8e
.f4~
.731
.828
.81l8
.962
.752
.A14
.e.,o
.9711
.6611
.753
.632
.92"
.636
.703
.765
.881
.592
.(,62
.7 .. 11
,1147
.5~5
.523
.626
.718
.820
... 00
.693
.797
... 73
.556
,e55
.76 11
.819
.e72
.7"3
.7.,7
,513
.579
.661
.755
.4&2
.534
.622
.720
."23
.501
.593
.695
.~73
.15c.7
.~t&tJ
.8l'5
.5....
.6n8
.687
.778
.'152
..... 2
.941
.'1112
... 43
.719
.1106
.371
.",\7
.II~3
.626
.685
.757
.841
.39"
... 76
.937
.67'!
.736
."05
... 76
.6 .. 2
... '11
.337
... 30
.53"
.""2
.812
.859
.916
.982
.735
.7114.
.G42
.911
.€o71
.722
.783
.854
.618
.€o70
.734
.809
.574
.536
.593
.6,:,2
.7 .. 3
.504
,563
.636
.71'!
."53
.383
.'156
.543
.3 .. 0
.312
.3"9
.495
.59"
.302
,39C
.400
.588
.25~
.5?':'
.623
.325
,"03
.503
.602
,3C3
.4~2
.37e
.477
.~37
.3 .. 0
... 21
.'i13
... 11
.293
,595
,el\3
."14
."(12
.565
,657
.5f3
.~7e
.806
.1l48
.1\99
.958
.729
.773
.P25
.665
.710
.76 ..
.€o12
.658
.715
.567
... 15
.7 .. 5
.405
... 6&
.'141
.6?5
.375
... 40
.,,18
."04
.350
."19
.501'
.5/19
.331
.403
... 87
.'177
.315
.390
.476
.5':'8
.291
.371
.u60
.553
.274
.762
.4 .. 5
.502
.572
.652
.281
.36"
.145'"
.1I2~
."97
.5 .. 9
,61"
.69n
.302
.3110
.8r.6
.529
.5/10
.642
.714
.~~~
.~~~
.eol
.839
.M5
.938
.724
.763
.CI0
.865
.660
.700
.606
.648
.699
.759
.'i61
.600;
.5?3
.5<,9
,293
.36"
.4"'i
.'130
.355
... 37
.523
.3"'7
.6'10
.30&
.374
.454
.538
.272
."26
."97
."7,,
.323
.3811
.465
,'1"1)
.2el
.6~5
.399
... 53
,521
.597
.343
.4C5
."7,)
.721
."39
.4119
,'\53
.625
.:!r,6
.1'.'56
.491
.538
.596
.66"
.264
.!f.ll
'''25
.">13
.486
.528
.581
.643
... 33
.479
,'i36
.':'O?
.393
..... 2
.503
.573
.299
.3<'1
, "3"
,512
,285
.27..
,3 4 0
."17
... 97
,
2,5
2" 1.0
1.5
.I\O~
.796
.832
.873
.921
.719
.755
.7~8
.847
.74')
.80':'
.655
.692
,736
.7811
.602
,640
,686
.740
.62~
.':'95
.773
.1'.75
.557
.515
.'\9':'
.~t,O
.644
.701
.610
.669
.516
.361
."1"
.1179
.1Ii\52
.4~6
.5~O
,3~1;
.3 0 2
.""0
.5;5
.316
• :a, 7 !")
,446
.523
.~68
.560
•
~~)O
,42"
.50"
." 31
.51e
.26~
.333
.410
,491
.~I)'
.:!~8
. . If'
.255
.326
,If';5
.4~(,
.....
~
TABLE 5.1.1 (Continued)
b = 1.5 a. = 0.05
al f
a2
1
2
5
II
5
&
7
8
9
10
12
111
16
18
20
22
2'+
26
28
30
.988
.962
.9113
.929
.918
.909
.902
.896
.891
.8!!7
.'''13
1 1.0
1.5
2.0
2.5
2 1:0
1.5
2.0
2.5
3 1.0
1.5
2.0
2.5
II 1.0
1.5
2.0
2.5
.989
5 1.0
1.5
2.0
2.5
.923
6 1.0
1.5
2.0
2.5
.967 '.875
7 1.0
1.5
2.0
2.5
.931
8 1.0
1.5
2.0
2.5
.9011
9 1.0
1.5
2.0
2.5
.e1'\2
1.01111
10 1.0
1.5
2.0
2.5
.988
.86'1
.969
.838
.810
.951
.787
.911
.769
.1l79
.923
.857
.807
.769
.938
.739
.887
.715
.846
.9 .. 3
.696
.A12
.9"'1
.9&7
.931
.9011
.882
.86'+
.837
.818
.803
.792
.783
.776
.769
.76"
.760
.756
.875
.838
.810
.7,.,7
.769
.7'+1
.720
.705
.6""
.991
.6811
.983
.677
.976
.670
.970
.665
.965
.660
.~57
.96~
.9~7
.807
.756
.95e
.716
.892
.68,\
.839
.661
.797
.9'1B
.6'11
.76;>
.898
.769
.716
.923
.676
.856
.739
.ge3
.715
.9,;2
.696
.9'+6
.667
.920
.6116
.901
.630
.tl87
.618
.876
.608
.1\£,7
.601
.A60
.59'+
.85"
.589
.1l"9
.58"
.ell5
.580
.685
.1l97
.661
.875
.6111
.858
.610
.631
.588
.812
.572
.796
.5~9
.5119
.777
.541
.770
.535
.764
.995
.52':1
.75'1
.9ge
.52"
•.71\6
.'120
.7'H
.'l83
.644
.82/\
.618
.806
.644
.802
.975
.611
.774
.584
.Q4'
.929
.618
.759
.91'1
.584
.72'1
.A87
.557
.7n6
.597
.723
.863
."i"3
."93
.A36
.985
.7~1
.8F-6
.5",5
.6<,8
.81'1
.964
.7~~
.96f
.~II1
.597
.788
.966
.566
.761
.963
.5113
.7111
• ':1115
.526
.726
.931
.513
.7t5
.920
.502
.706
.912
."911
.696
.905
.'+87
.692
.899
... 81
.687
.89 ..
.476
.682
.890
."72
.f7S
.563
.732
.912
.~30
.70"
./\86
.506
.68"
.867
... 89
.669
.1'\53
.475
.6'17
.8 .. 2'
... 65
..... 7
.A33
."56
.6.. 0
.1l26
..... 9
.633
.A20
..... 3
.628
.A15
."'/\
.623
."33
.619
.e07
.995
.501
.657
.e22
.990
.477
.637
.P03
.972
.459
.<';>1
.7AA
.958
.445
... 33
.,\99
.766
.'139
.4211
.591
.761
.932
.417
.5115
.755
.926
... 11
.57'1
.750
.921
."06
.575
.7"5
.917
."01
.609
.777
.9 .. 7
.'177
.619
.769
.922
.452
.597
.749
.90'+
... 33
."illl
.73'+
.1\90
.'119
.51'.9
.723
• 879
... 08
.'1'19
.391
.3B"
.37"
.54 4
.~311
.53\6
.71~
.39A
.551
.706
.BE>3
.695
.852
.690
.A70
.700
• 857
.37"
.&30
.667
.535
.6A7
.1\ .. 8
.512
.6"9
.796
.9"8
.All
.99'1
.8"P
.~~6
.~71
.7~2
.913
.e"..
.....
N
TABLE 5.1.1 (Continued)
b = 1.5 ex = 0.05
al f
a2
2
3
12 1.0
1.5
2.0
2,.5
.962
.837
.922
1/1 1.0
1.5
2.0
2.5
.9"3
1
16 1.0
1.5
2.0
2.5
18 1.0
1.~
.929
.9l)7
2.0
2.5
20 1.0
1.5
2.0
2.5
.909
.95"
22 1.0
1.5
2.0
2.'5
• 902
.9'12
.9117
2'1 1. a
1,5
2.0
2.5
.896
.933
16
18
20
22
211
26
.11'11
.558
.685
.!l15
."1'1
.535
.66'1
.796
.395
.518
.6'19
.71\2
.379
.51)5
.637
.771
.367
... q!\
... 27
.3'\7
... /17
.620
.75'1
.3"9
.'11'0
.613
.859
.1177
.590
.713
.8"2
.3'13
.117"
.606
.7" 3
.'152
.5 .. 7
.652
.763
."1"
.51"
.366
." 72
.51lS
.700
.337
.""7
.327
.319
,432
.4~1
.572
.6·9
.f)~2
.~~4
.735
.387
."90
.601
.715
.3"i0
.6'16
.1\0 3
.'177
.5<,9
.672
.7nl
.679
... 72
... 6'>
.5"~
.He
.'189
.'165
.651
.7'1'1
.t+t!\9
."33
.'115
.605
.702
.395
.'190
.57'1
.673
.36&
.'155
.552
.652
.3.. 5
.'136
.'135
.f>37
.3211
."22
.5;>2
.6;>5
.315
.'111
.30'1
."C'1
.'103
.608
.295
.39"
.497
.601
.21:18
.387
.626
.721
.'1 .. 5
.513
.'119
.'56'1
.65"
.328
.1108
... '1'1
.5116
.297
• 3e 1
."71
.2e6
.372
.5 0
.350
."28
.513
... 02
.311
.3 0 3
.61".697
.379
."54
.536
.1'02'1
.• 574
• ~6'+
.502
.558
.621
.691
.'165
.'1;>2
.'1111l
.660
.'133
.'193
.51'01
.'1011
.'169
.539
.615
.367
.'133
.506
.511'1
.337
.'106
.1182
.562
.315
.3F1&
."6'1
.5 .. 5
.2'17
.370
.'I"iO
.533
.283
... 911
.5'13
.600
... 62
.'156
.'107
.'1,"
.U78
.'<>"6
.~"7
.. 3~1
.5~1I
.327
.38/l
... 56
.:'I~"
• ~C;(I
... 37
.630
.6r'l
.398
... 53
.'11'"
.5/!3
.357
."1F,
."1\1
.75"
.5'11
.58'1
.6"2
.70:>
.~51
.~211
.670
.710
.7511
.1'02
.5'111
.635
.681
.732
.535
.577
.625
... 7"
.'187
.531
.582
.637
..... 9
... 9'1
.5 .. 7
.605
.'117
.'1"'5
.519
.578
.391
..... 0
.1196
.557
.3'19
.'101
."60
.52"
."11
.3'13
7
8
.7111
.1I30
.933
.667
.761
.869
.986
.~10
.70"
./121
.9'11
.566
.668
.78'1
.907
.530
.636
.75'\
.Ml
.6"6
.72'1
.C1"
.913
.588
.670
.76'1
.867
.5.. 3
.629
.726
.1l31
.50G
.705
.769
.8 .. 2
.922
.630
.6')7
.773
.858
.5 7 2
.6'12
.722
.610
.526
.599
.682
.773
2
.,. .. 5
.90"
.971
.6':1'1
.7"9
.812
.881
.618
.676
.7"2
.Fl15
.55'1
.620
... 8'1
.76F,
.513
.576
.6"8
.727
.763
.830
.6a'l
.733
.Te8
.8'19
.608
.659
.717
.782
• 5'19
.603
.663
.731
.601
.6'16
.915
.677
.720
,i(-,9
.11:>3
.759
.807
.849
.8"5
.l!1)3
.6FII
.7 0
.1\1'2
.9 .. 0
.~7,
1_
(,
.9~3
.918
.969
12
5
.818
.f!.1'9
.970
.776
.817
.e~"
,
10
"
.720
.795
.881
.')76
.#;97
."qS
.1175
.'1 .. 1
.5~1
.611
.7~2
.5~7
0
.673
.~~S
.4n~
.F.22
.&lee
.7"~
.'il?
74~
.1416
.0,33
.6~h
.';~2
.2P2
.276
.37 0
."€2
.'187
.3,c:.?
."~"
.~~~
.44
.550
.'5'4'+
.53Q
.'\3'1
.439
.'122
.271
.3 .. 8
.'130
.'11"
.262
.3" 0
."23
.507
.25 ..
.333
.'117
.502
.2"7
.3;>7
.'111
."97
.;>"2
.322
.'-07
."93
.260
.3;>1J
... 03
.250
.;>2'1
.3~~
.~79
• ~11
.U~A
.4.0.8
.47Q
... 72
.2"2
.313
• ~,89
... 67
• 2 3£.
• ~t, 7
.'I2~
.:>71
.33'1
... 12
.""'2
."'\7
.319
.373
... 35
.501
.;>95
.352
."16
.41\3
.277
.;>62
.323
.;>50
.31;>
.231
.291>
.2~t;,
.290
.~~~
... 0 1
.46'1
.~89
.~80
.~65
.3~r:
."59
."50
.2 .. 0
.303
.372
..... 3
."37
."3;>
.355
.";>7
.312
.361
... 17
.'177
.288
.33'1
.269
.3?3
.2~t.J
.231
.2 P Q
.209
.27C
.~q7
.3~2
... 45
.'13'1
.'118
.223
.2"'2
.3"&
.'111
.27~
.'159
.2"2
.2"1J
.360
.'125
.21£'
.309
..... 3
.501
.3 .. n
.'106
... 02
.247
.:>911
.;>35
.;>87
• 3~ 4
• ~'I"
.21&
.270
.32e
.369
.20P
.263
.323
.38"
.;>02
.!.A2
.263
.311
.31;6
.u ..
.209
.25'1
.202
.253
.1'15
.;>"7
.!O3
.3&0
.3~6
.~'I8
.32::
.~q~
.529
.567
.511
.659
."81
.521
.567
.617
... 43
.532
.56'1
.557
.38"
."29
... 79
.535
28 1.0
1.5
2.0
2.5
.887
.917
.951
.f>('0
.693
.730
.769
.511'1
.618
.656
.699
.52'1
.559
.59'1
.6'13
.'176
.513
.55"
... 00
.1131'
.1175
.519
.'1 .. 6
.'Ir6
.'1 .. 5
.'1119
.539
.379
.'119
.'1&0;
.51"
.337
.380
.'129
... 82
.30(,
.351
.'102
."57
.2112
.3;>9
.981;
.71-0
.7°2
.826
.86'1
... 38
.u,:?4
.'112
.403
.22'1
.277
.335
.396
30 1.0
1.'5
2.0
2.'5
.8113
.912
.14;>
.975
.756
.7rS
.817
.8·2
... 57
.687
.51'0
.611
.6 .. 1;
.6P'I
.5;>0
.55:?
.51.'9
.629
.'172
.5C6
.5'1'1
.585
.'1B
.1I6F1
.'\07
.551
.'I~1
.37"
."11
.115 4
.500
.33;>
.371
.416
.'165
.301
.3"2
.3119
.'139
.n6
.319
.3"8
.'120
.;>57
.302
.3·.2
.'106
.2'12
.288
.229
.;>77
.329
.3115
.;>18
.267
.3;>1
.377
."76
.523
.301
.~2~
.537
.SSf
.589
.£,2&
.1'067
.713
.756
.306
."6~
.519
.;>63
.665
.701
.7111
.7/1'1
.'I~7
.7~S
.269
.3'\7
.450
.277
.76"
.7 e1 9
.6:'17
.878
.7~O
.60!
.739
.3""
."'16
.891
.'12'1
.9(,1
1.000
."s"
.5n3
... 91
..... "
.591
26 1.0
1.5
2.0
2.5
... e'l
.312
.425
.5 4 2
30
.332
.50;<,
."~;>
.615
•
26
.337
• ~'90
."~ 70
•.'\V?
.39'1
.~c)2
.~7C
.3~~
c
.3t)j:l
.365
.257
.3"7
• 144
1
.!~2
.211J
.~3'1
.2~'!
.318
.3PO
......
W
TABLE 5.1.1 (Continued)
b = 2.0 a = 0.01
al f
1
2
8
5
6
7
"
1.717 1.628 1.55" 1."93 1 ..... 1 t.39A
II
,
a2
10
12
1"
16
18
20
22
2-
26
28
30
1 1.0
1.5
2.0
2.5
1.'28
2."11
2.891
3.385
1.821
1.3~2 1.331 1.281 1.2_2 1.212 1.1~7 1.161
2.2'5 2.107 2.0'6 2.022 1.961 1.'11 1.A6' 1.83_ 1.S0" 1.756 1.71' 1.~'0 1.641 1.6"8
2.17.. 2.662 2.511 2.491 ~ ... 37 2.361 ~.~"1 2.312 2.283 '-.231 2.2~1 2.11" 2.1~2 ~.135
3.256 3.1"2 3.050 2.916 2.'11 2.868 2.82A 2.1 a 5 2.167 2.722 2.68~ 2.661 2.~40 2.62"
1.1~0 1.136 1.12.. l.ll" 1.105
1.~33 1.620 1.609 1.5~' 1.~91
~.120 2.10' ~.097 2.0pP 2.0fl
2.~10 2~598 2.588 2.~.0 2.~72
2 1.0
1.5
2.0
2.5
1.821
2.265
2.715
3.163
1.695
2.126
2.566
3.013
1.273 1.239 1.210 1.165 1.131 1.105 1.0p" 1.06~
1.106 1.6~5 1.6"9 1.608 1.578 1.~55 1.5'8 1.52"
2.153 2.12" 2.100 2.063 2.036 2.016 2.000 1.~88
2.601 2.560 2.55R '.524 2.500 2."01 2.467 2.455
1.054 1.043 1.03.. 1.0~o 1.0~O
1.513 1.~0" 1.496 1.4q~ 1.484
1.978 1.970 1.~~3 1.9~7 1.9~2
2.44E 2.438 2.432 2.~26 2.4~1
:5 1.0
1.5
2.0
2.5
1.711 1.535 1.415
2.116 1. 9 70 1.8~5
2.523 2.368 2.250
2.93~ 2.773 2.656
1.385
1.765
2.162
2.510
1.313
1.695
2.095
2.505
" 1.0
1.5
2.0
2.5
1.628
1.986
2.351
2.722
1.296
1.63"
1.991
2.360
1.224 1.166 1.119 1.080
1.565 1.511 1."~6 1.434
1.926 1.e7~ 1.B36 1.A05
2.291 2.250 2.213 2.1A3
5 1.0
1.554 1.422 1.313 1.22" 1.152 1.094 1.041
1.875 1.72A 1.613 1.525 1.~~6 1.~03 1.360
2.20 4 2.0~8 1.9~2 1.~46 1.101 1.731 1.692
2.539 2.318 2.263 2.179 2.111 2.069 2.033
1.009 .971 .950 .908 .818 .85" .637 .~22 .8U .801 .19,5 .1e6 .7eo
1.326 1.298 1."5 1.239 1.'13 1.19" 1.178 1.166 1.156 1.141 1.140 1.134 1.129
1.661 1.635 1.61" 1.~e2 1.55A 1.540 1.5'6 1.514 1.505 1.4~1 1.490 1.~p~ 1.4eO
2.003 1.979 1.959 1.929 1.906 1.AA9 1.B75 1.~65 1.956 1.B48 1.8"2 1.836 1.831
6 1.0
1.5
2.0
2.5
1.493 1.362 1.25"
1.iA' 1.636 1.5'3
2.0~0 1.925 1.810
2.384 2.224 2.109
1.23~ 1.2~1
1.5~8 1.512
1.84~ 1.824
7 1.0
1.5
2.0
2.5
1.441
1.704
1.916
2.253
e 1.0
1.5
2.0
2.5
1.3 a 8 1.273 1.161 1.080 1.009 .950 .902 .AE' .820 .aoo .75f, .724 .699
1.639 1.496 1.30" 1.2 9 7 1.228 1.173 1.129 1.093 1.06" 1.040 1.000 .916 .955
1.·e7 1.735 1.621 1.~35 1.4(9 1.417 1.377 1.345 1.318 1.296 1.2&3 1.23A 1.'19
2.141 1.ge3 1.869 1.784 1.721 1.613 1.635 1.~04 1.5aO 1.559 1.528 1.505 1.4A1
, 1.0
1.5
2.0
2.'5
1.362 1.239 1.135 1.048 .971 .918 .869 .1\28 .79" .166 .121 .681l .662 .643 .627 .615 .604 .596 .5.' .582
1.~~2 1.442 1.331 1.24~ 1.174 1.119 1.07" 1.039 1.0~9
.91\4 .946 .918 .891 .pal .A61 .P57 .348 .8~0 .83'1 .829
1.~11 1.~&1 1.541 1.4~1 1.3 0 4 1.342 1.301 1.~69 1.242 1.220 1.185 1.160 1.1~1 1."6 1.114 1.10~ 1.0'S 1.~89 1.0~3 1.:7.
2.~45 1.~S9 1.774 1.&90 1.~26 1.511 1.538 1.50~ 1. 4 03 1.4~2 1.430 1.40f, 1.,p8 1.37" 1.~63 1.35" ~.346 1.'3~ 1.3~" 1.~29
10 1, 0
1.5
2.0
2.5
1.331 1.210 1.1r1 1.0?1 .9!OO .1\90 .e"l .AOO .766 .731 .691 .657 .631 .611 .595 .582 .571 .~62 .~5S .'54~
1.53" 1.396 1.2~6 1.1'9 1.12'3 1.073 1.028 .991 .9<;1 .93'" .e97 .C6Q .1\~1 .1\'0 .A17 .806 .1 a 7 .769 .783 .777
1.146 1.~97 1.404 1.397 1.330 1.'-1~ 1.236 1.203 1.176 1.153 1.118 1.093 1.073 1.05P 1.0"6 1.G36 1.0~5 1.021 1.Gl~ 1.0GQ
1.9E2 1.801 1.G a 3 1.600 1.54~ 1.49~ 1.455 t.42" 1.399 1.370 1.345 1.321 1.303 1.209 1.271 1.26A 1.260 1.253 1.24A 1.2"3
1.~
2.0
2.5
1.495
1.838
2.195
2.560
1.313
1.506
1.882
2.094
1.585 1."95 1."22 1.362 1.313
2.011 1.9~0 1.04A 1.790 1.7....
2."49 2.358 2.2~8 2.232 2.188
2.09" 2.fO" 2.136 2.683 2.6"1
1.385
1.123
2.018
2.44"
1.206
1.4'"
1.706
1.979
1.254
1.6"0
2.0"3
2 ... 51
1.206 1.161 1.135 1.107 1.065
1.596 1.~61 1.532 1.50A 1.411
2.003 1.971 1.945 1.923 1.890
2.419 2.3A9 2.364 2.3 .... 2.'13
-1.166 1.094 1.036 .989
1."35 1.~66 1.312 1.2E9
1.72" 1.659 1.609 1.569
2.025 1.963 1.915 1.818
.950
1.048
1."06
1.780
2.160
.'318
1.033
1."45
1.866
2.291
1.010 .9 9 1 .'H7 .965 .9~5 .947 .940 .«:lI!4
1.4'5 1.4n9 1.'91 1.386 1.378 1.371 1.3~5 1.359
1.q48 1.8'3 1.A22 l.el2 1.80" 1.790 1.7 9 ) 1.7e7
2.'73 2.260 2.'4' 2.239 2.232 2.225 2.2~C 2.215
1.021 .980 .949 .926 .908 .99" .883 .1113 .865 .a~e .8S3 .
1.363 1.347 1.321 1.302 1.2~6 1.,7~ 1.26~ 1.256 1.24p 1.24~ 1.237
1.159 1.126 1.102 1.~A" 1.670 1.659 1.6~9 1.641 1.~35 1.~29 1.~24
2.1"0 2.109 2.067 2.070 2.0~6 2.045 2.036 2.029 2.322 2.017 2.012
.890 .848 .817 .79&4 .776 .761 .1 .. 9 .740 .131 .125 .119
1.183 1.1'" 1.121 1.101 1.0A5 1.n73 1.063 1.05~ 1.041 1.041 1.036
1."91 1.'158 1."3" 1.416 1."~2 1.~90 1.3~1 1.373 1.366 1.3~~ 1.3~~
1.005 1.114 1.151 1.734 1.1'0 1.709 1.700 1.692 1.6B6 1.6~n 1.67E
1.119 1.047 .989 .9 .. 1 .902 .869 .841 .198 .167 .143 .12" .109 .697 .681 .679 .6 7 2 .f6€l
1.360 1.291 1.23~ 1.193 1.158 1.130 1.106 1.069 1.0"2 1.022 1.006 .994 .983 .975 .9E7 .c?t;1 .q~6
1.f21 1.5~6 1.505 1.465 1.43" 1."08 1.3b6 1.353 1.328 1.310 1.2~5 1.28 4 1.274 1.266 1.,!~ 1.2~4 1.2~9
1.A95 1.832 1.784 1.741 1.111 1.693 1.612 1.~41 1.618 1.601 1.sn1 1.516 1.561 1.~~3 1.553 1.547 1.5'"
.6 A O
.~:l.9
1.~04
1.~73
.~65
.653 .6 .. 3 .634 .627 .e-21
.926 .915 .907 .A99 .e9~ .e~9
1.193 1.1A3 1.175 1.168 1.162 1.151
1.462 1.452 1."45 1.436 1.~33 1.428
....,
~
TABLE 5.1.1 (Continued)
b = 2.0 a = 0.01
al
f
1
:2
3
II
5
6
7
8
,
az
1'1
16
18
20
12 1.0
1.5
2.0
2.5
1.165 1.065 .'80 .'08 .8'18 .7'8 .756 .721 .6'1 .6'13 .608
1.'156 1.323 1.21'1 1.127 1.057 1.000 .'5'1 .916 .8e5 .856 .816 .788
1.63'1 1.4''1 1.3A2 1.2'5 1.227 1.174 1.131 1.0'16 1.0~8 1.0~5 1.009 .982
1.~2? 1.~75 1.5~1 1.476 1.'110 1.360 1.320 1.267 1.2"'1 1.''10 '.~06 1.161
.!lSO
.765
.9"'2
.5,\'
.542
.734
1'+ 1.0
1.5
::>.0
1.2'12 1.1:'11 1.0:'13 .9~9 .878 .817 .71:.7 .72'1
.6118 .657 .f.08 .571 .'1'12 .5'0 .0;02 .~"8 .~76 .4~6
1.:395 1.266 1.1<,0 1.073 1.003
.9'15 .898 .65'" .826 .799 .757 .726 .702 .61''1 .~69 .6'\8 .6148 .64Q
1."53 1.~1! 1.3r'l 1.217 1.1"9 1.09~ 1.050 1.Cl'1 .9a5 .%1 .923 .696 • A75 .A<;9 .D~E) .635 .D;;t6 .1'19
1.722 1.573 1.'I~0 1.37'1 1.307 1.255 1.21'1 1.181 1.15'1 1.132 1.097 1.072 1.053 1.0'7 1.025 1.016 1.007 1.000
2.~
III
12
1.2~1
1.212 1.10~ 1.010 .926 .85'1 .79'1 .743 .(,99 .61'2 .631 .580
1.!>'IP 1.22~ 1.116 1.032 .%1 .9C3 .e·55 .1115 .71\1 .753 .709
1.49/l 1.35' 1.2"'1 1.1'\7 1.087 I.O~:> .987 .'150 .9,0 .89'\ •• '16
1.638 1.'192 1.3ro 1.293 i.226 1.173 1.131 1.0'17 1.0<,9 1.0"6 '.010
.5'12
.677
.1127
.98'1
.513
.€-52
./I/lE.
.9(,'1
18 1.0
1.5
2.0
2.'5
1.1/17
1.3C9
1.'137
1.'570
.'IA9
20 1.0
1.5
2.0
2.5
1.167 1.068 .977 .89'1 .6:>2
1.277 1.159 1.0~9 .97'1 ."03
1,~93 1.263 1.1~6 1.070
.99'1
1.«13 1.374 1.26Q 1.111 1.106
22 1.0
1.5
2.0
2.5
.761
.8'1'1
.'42
.611
.717
.8'11
.97(,
.559
.672
.1'01
.'3'1
.520
.638
.771
.91:>
.7'19
.1''12
.6,7
.717
.825
.944
.595
.688
.798
.919
.5'12
.1'.'11
.756
.1\81
.502
.60t.
.72t.
.853
.'171
.'11'0
.702
.1\32
.615
.6'1'1
.7'10
.6'17
.562
.664
.762
.871
.529
."'16
.719
.fl32
.'188
.580
.688
.803
.'156
.6~'I
.665
1.05~
.709
.794
.895
1.C09
1.150
1.250
1.3'56
1.466
1.054 .965 .883 .611 .7'19
1.136 1.0~8 .95'1 .~!!2 .822
1.22'1 1.1''1 1.038 .966 .910
1.329 1.220 1.13'1 1.06'1 1.009
.697
.772
.862
.96'1.
.653
.730
2_ 1.0
1.5
2.0
2.5
1.136
1.228
1.325
1.426
1.043 .C:;~5 .873 .eOl
1.117 1.0'0 .937 .A(,5
1,2Cl 1.0'18 1.012 .941
1.292 1.18'1 1.098 1.G28
.7'10
.805
.A83
.972
.687
.75'1
.A34
.926
.<''13
.712
.79'1
.1188
.675
.7(.1
.6.,7
.571
.6'15
.733
.631
.517
.595
.<'1111
.791
26 1.0
1.5
2.0
2.5
1.12'1
1.2C9
1.298
1.391
1.034 .~47 .865
1.101 1.006 .923
1.171 1.075 .9.,0
1.260 1.15'1 1.068
.793
.1\51
.919
.998
.731
.790
.660
.9'10
.679
.739
.Al1
.119'1
.(,3'1
.(,96
.771
.1156
.5"'6
.61'>0
.736
.82'1
.562
.626
.708
.797
28 1.0
1.5
2.0
2.5
1.11'1
1.1'12
1.'75
1.31'2
1.026
1.087
t.157
1.233
.940 .658
.9"3 .911
1.0'16 .972
1.128 1.0112
.786
.639
.901
.972
.7:>5
.778
.8'11
.91'1
.672
.727
.792
.A67
."'27
.51\8
.646
.716
.795
30 1.0
1.'5
2.0
2.'5
1.105
1.177
1.255
1.335
1.020 .934 .1\53
1.07'5 .9113 .900
1.B9 1.0'10 .9%
1.210 1.lC6 1.020
.780
.1l29
.8115
.9'19
.71'1
.7611
.625
.A91
.666
.716
.775
.8'13
.51\2
.6'5
.6')8
.771
<
.70;~
.1157
.973
.1123
.9:>7
.1'.83
.751
.~~A
.1'.21
.672
.733
.110'1
.9~6
1.1~3 1.1~8
16 1.0
1.5
2.0
2.5
1.06'1 .9"1 .9/\6 .637 .776 .72'1 .hAO .6'13
1.l/1A 1.0~6 1.0nO .Q?9 .67/\ .621 .760 • 7~6
1.303 1.1"5 1.109 1.039 .962 .936 .899 .867
1.'127 1.316 1.229 1.1"'1 1.107 1.06'1 1.029 1.000
.7~8
.~12
.'189
."'33
.7"Q
.9"9
.'1"5
.5'"
.732
.87~
22
26
26
3C
.52' .517 .508
.723 .713 .705
.'13" .923 .915 .9~S,
1.136 1.127 1.119 1.11'
.500
.l.9'l
.9C;>
.4'4
.471
.<,111
.77'5
.936
.4'16
... 77
.718
.11(,3
2"
.9«; ..
.~O7
.9iot Q
.ld3
.!J96
.-:£7
.~2"
.~~~
.76'1
.926
.70;5
.916
.7~~
.741
.!?:.
.B('
.'11
.%5
.1\99
.'130
."17
.~17
.-;C.3
c~9r:!
.S~7
.697
."07
.0;'14
.689
.3'17
.!.64
.707
.1153
• (,83
.'531
.677
.P4'1
,°37
•P~1
.~~.,
.396
.516
.6'19
.733
.385
.37~
.367
.5~9
.5Cl
.l.6q~
.641
.776
.63!4
,7;Q
.62 0 .76 ..
.3~7
.349
.471
.593
.717
.465
.816
.f'!)3
.782
.'1'0
• 5 ~1
.645
.7(,5
.'110
.0;1'1
."'30
.751
.393
.500
.618
.7'10
.379
.'189
.606
.731
.367
.'179
.600
.72'1
.'176
.558
.656
.761
.4'13
.0;30
.631
.739
.4t7
.5n8
.611
.722
.~96
.490
.0;36
.708
.379
.47(,
.696
.36'1
.'11'>'1
.574
.687
.352
.'154
.565
.679
.506
.578
.f.62
.756
.'166
.5'10
.628
.72'5
.'133
.'Itl
.F.03
.702
.4n7
.4.ne
.367
.455
.55'1
.659
.352
.6A4
.385
.'170
.0;67
.1',70
.6 4 9
.555
.615
.686
.766
.500
.56'1
.f.40
.72'5
.'158
.525
.605
.694
.42'1
.'195
.579
.1'.70
.317
.'171
.50;6
.6'12
.~75
... 53
.'\112
.1',37
.357
.'137
.529
.626
.5118
.603
.6(,6
.7113
.49'1
.551
.1.21
.f.9'1
.'151
.512
.585
.667
.'117
.'11\1
.556
.1','13
.390
.'1<;7
.537
.6''1
.'67
.1138
.')20
.t.09
.349
.422
.506
.597
.5P3
.'=, ..
'''~1
.(,2 7
.443
.61'4
.f (,'1
.4!A
.633
.60;
.'110
.5:>9
.656
.792
.~53
.";"-
.~56
.'126
.0;'12
• .;70
.446
.5'1'1
.~'3
1.1~' 1.1~1
.~8'1
.~!7
.712
.3"2
.333
.4 .. ;
.673
.'139
.'\'52
.667
.3"0
... 33
.535
.6"1
.329
.'12'1
.528
.63'1
.320
• "1 7
.521
.628
.3 4 2
.'12'5
.518
.616
.329
.414
.509
.31 0
.405
.309
.5~1
.60£1
.6Cl
.'195
.59'1
.333
.'109
.'195
.587
.320
.398
.'Ie6
.0;78
.30'1
.389
.'178
.571
.443
.5 4 '+
.~5~
.!9!
.29"
.3~1
... 71
• c.~~
--.J
VI
TABLE 5.1.1 (Continued)
b = 2.0 a = 0.05
al f
1
2
22
2'1
1.560 1.'167 1.3" 1.3'10 1.2" \.~5~ 1.2~0 \.250 \.166 1.137 1.115 1.0~8 1.084
2.011 1.9'0 1.850 1.797 1.755 1.721 1.6~3 1.~70 1.63~ 1.609 1.~A9 1.573 1.~61
2.472 2.382 2.315 2.263 2.2~3 2.1°1 2.1~5 2.143 2.110 2.086 2.a~8 2.0~4 2.0 4 3
2.939 2.6~0 2.7115 2.735 2.~96 2.~~6 2.641 2.621 2.590 2.568 2.551 2.5~8 2."2e
1.073
1.551
2.034
2.S20
1.06"
1.543
2.02 7
2.513
II
5
1.832 1.681
2.2~~ 2.134
2.7i2 2.~95
3.233 3.C62
2 1.0
1.5
2.0
2.5
1.661 1.524 1.404
2.0~2 1.~15 1.795
2.4CJl 2.320 2.2Cl
2.906 2.733 2.617
3 1."
1.5
2.0
2.5
1.Sf1 1.404 1.2115 1.196 1.1~9
1.903 1.73~ 1.620 1.535 1.473
2.257 2.08~ 1.972 1.691 1.1133
2.617 2.446 2.334 2.256 2.1CJ9
1.0
1.5
2.ll
2.5
a2
20
3
1 1.0
1.5
2.0
2,5
'I
,
6
7
8
10
12
1'1
16
111
26
28
30
1.056 1.049 1.04.
1.536 1.531 1.';26
2.021 2.016 2.~12
2.508 2.~C! 2.'CO
1.314 1.245 1.193 1.151 1.116 1.0~1 1.069 1.035 1.011 .<]')2 .976 .967 .957 .<]50 .<]43 .'n7 .Q!3
1.708 1.644 1.596 1.559 1.529 1.5n6 1.407 1."57 1.43~ 1.41') 1.406 1.~95 1.3~7 1.379 1.373 1.36~ 1,3~3
2.118 2.0~6 2.014 1.979 1.q~2 1.9~1 1.913 t,P65 1.664 1.~46 1.6~6 1.~26 1.n17 1.Al0 1.P.04 1.7'19 1.7'15
2.S37 2.481 2.43CJ 2.406 2.'AO 2.3~9 2.342 2.315 2.295 2.279 2.2~7 2.257 2.249 2.242 2.236 2.231 2.~26
1.077
1.426
1.789
2.157
1.036 1.004 .978 .956 .n3 .e99 .Illli
.A66 .A55 .845 .637 .~~1 .P.o2 c
.::;~~
1.390 1.'61 1.3'8 1.319 1.21l9 1.'67 1.?50 1.237 1.?26 1.217 1.210 1.?C4 1.19" 1.1<;4
1.755 1.72n 1.7r6 1.66n 1.660 :.639 1.623 1.6tO 1.~00 1.S91 1.584 1.:78 1.573 1.'68
2.125 2.0~~ 2.077 2.0;0 2.033 2.012 1.9'17 1.9~4 1.974 1.966 1.9~9 1.953 1.94P 1.9~~
1.467 1.314 1.1°6 1.108 1.040 .951\ .947 .Q14 .8A6 .666 .832 .607 .71\9 .774 .762 .752 .744 .738
1.7f4 1.600 1.4~3 1.399 1.337 1.289 1.253 1.2~3 1.2~0 \.lAO 1.150 1.127 1.110 1.097 1.01\6 1.071 1.070 1.G6~
2.r70 1.903 1.7~P, 1.708 1.6h9 1.604 1.S70 1.542 1.520 1.502 1.473 1.452 1.436 1.423 1.412 1.4r4 1.~"7 1.391
2.~84 2.215 2.102 2.024 1.967 1.925 1.892 1.A~5 1.8'14 1.826 1.799 1.77A !.7~2 1.7~O 1.740 1.731 1.72 4 1.718
.73~
.727
1.G"~ 1.~53
1.~~~
1.~~1
1.71~
1.70Q
5 1.0
1.5
2.0
2.5
1.396
1.654
1.923
2.1 9 9
1.245
1. 4 93
1.75/\
2.'31
1.123 1.040 .972 .919 .878 .844 .617 .7~5 .760 .735 .715 .700 .6l18 .E.76 .6 7 0 .663 .&57 .6~2
1.317 1.2~2 1.229 1.1~1 1.143 1.114 1.0~9 1.06~ 1.03~ 1.016 .998 • Qll.5 .'17l4 .Q65 4951 .~50 • '1"t~, .Q40
1.6 4 3 1.5~2 1.50.'- 1.457 1.422 1.394 1.371 1.353 1.3l3 1.302 1.2A5 1.'72 1.262 1.253 1.246 1.24C 1.23 4 1.230
1.918 1.~39 1.7112 1.739 1.705 1.~79 1.6~7 1.639 1.611 1.590 1.574 1.'i~2 1.<;'i1 1.543 1.536 1.530 1.'52 4 1.C:2a
6 1.0
1.5
2.0
2.5
1.340
1.568
1.1\0'
2.C51
1.193
1.409
1.64'1.e"'4
1.077 ,968 .919 .665 .623 .789 .7F.l .736 .702 .676 .656 .641 .1;29 .08 .610 .£.03 .597 .591
1.2"3 1.207 1.144 1.095 1.056 1.026 1.001 .981 .949 .926 .906 ,894 .1\~3 .874 .666 .1\59 ,854 .P49
1.527 1.445 1.3/15 1.339 1.303 1.'75 1.252 1.233 1.203 1.181 1.164 1.1'11 1.1"0 1.132 1.12 4 1.118 1.11~ 1.1C~
1.771 1.6CJl 1.633 1.589 1.555 1.~2'" 1.5~6 1.4,.,1\ 1.459 1.43/1 1.422 1.409 1.399 1.391 1.31\3 1.377 1.372 1.!67
7 1.0
1.5
2.0
2.5
1.29S 1.151 1.036
1.711 1.~49 1.4~4
1.930 1.765 1.6~1
8 1.0
1.5
2.0
2.5
1.259
1.441
1.633
1.831
1.11/1 1.00 4 .91 4 .8 4 4 .789 .745 .709 .6.0 .656 .616 .591 • ";70
.5'13 .540 .530 .521
.513
1.2A7 1.172 1.0/\5 1.019 .969 .929 .AQ1 .871 .850 .017 .79'- .774 .759 .748 .738 .730 .723
1.47? 1.3~7 1.274 1.?12 1.164 1.127 1.0~~ 1.074 1.054 1.023 1.001 .9R3 .970, .959 .950 .942 .936
1.667 1.5-3 1.472 1.412 1.367 1.332 1.'104 1.2B1 1.?62 1.233 1.?1~ 1.195 1.1"2 1.171 1.163 1.155 1.149
'} 1.0
1.5
2.0
2.5
1.230
1.394
1.563
1.749
1.091 .978 .e6~ .1I17 .761
.716 .6/10 .6'i0
.~26
.587 .559 .!i37 .5'1 ,"07
1.242 1.127 1.~40 .974 .922 .8112
.1\4~
.8~3
./l01
.767 .742 .723 .7eA .~96
9
1.409 1.2 4 1.210 1.147 1.099 1.C61 1.031 1.007 .987 .955 .932 .915 .901 .1'90
9
1.51\5 1.471 1;390 1.329 1.283 1.240 '.21 1.196 1.177 1.l47 1.125 1.101l 1.0 9 <; 1.085
10 1.0
1.5
2.0
2.5
1.4~8 1.3~2
1.2~6
,c'i'+
;947 .e76 .62'1 .760 .745 .716 .693 .656 .629 .1>09 .5°3 .'i80
.546
.570 .%1
.':~2
1.1qO 1.075 1.02"; .986 .955 .9~O .'109 .A77 .653 .'135 .620 .A'J9 .1\00 .792 .79~ .7H .774
1.351' 1.290 1.24] 1.207 1.17A 1.1r;4 1.13~ 1.105 1.062 1.0~5 1.n~2 1.~41 1.032 1.025 1.018 l.al~ 1.COP,
1.571 1.512 1.46/\ 1.433 1.'105 1.363 1.364 1.336 1.314 1.?96 1.2 0 5 1.?75 1.266 1.259 1.253 1.247 1.243
.496
.f.R7
./180
.467
.676
.673
1.07~ 1.0~1I
1.250 1.069 .9~6 .666 .795 .736
.693 .656 .62 6 .600 .561
.532 .510 .4"3 .479 .468
1.3~5 1.20" 1.0~0 1.003
.936 .,.,O~ .042 .fl09 .71'2 .759 .72'1 .699 .LAO .6L5 .<;';3 .643
1.514 1.356 1.241 1.1'16 1.093 1.044 1.006 .97"> .9'\0 .929 .890 .874 .1'56 .B~2 .1'31
.822
1.679 1.517 1.403 1.320 1.259 1.213 1.176 1.147 1.1'4 1.104 1.074 1.052 1.035 1.0'1 1,011 1.002
.'+59
.f,35
.1l14
.994
.507
.71A
.93C
1.1"~
.479
.;72
.R66
1.062
1.0~7
.451
.h28
.807
.988
.444
.622
.£<02
.91'2
.!Gl
.713
.92!i
1.139
.47~
."1.,7
.6f>6
.861
.'f,1
.oS6
l.e"2
.436
."'1~
.797
.976
'oJ
0'\
TABLE 5.1.1 (Continued)
b = 2.0 ex = 0.05
al f
a2
1
II
II
9
10
12
111
16
18
20
22
211
26
2R
30
.5/17
1.035 .923 .832 .760 .702 .656 .618
1.1117 1.033
.9115
.876 .1523 .7150
.718
.7116
1.273 1.1~8 1.072 1.007 .957
.060
.~17
.R1l6
0
1.110'1 1.2 11 1.210 1.11&8 1.100 1.063 1.033 1.0n8
.561
.6911
.839
.'10/1
.520
.651l
.P05
.'157
.490
.632
.781
.93'1
.467
.612
.7(,3
.'n7
.41&"
.5'16
.7'1/1
.903
.'135
.'583
.737
.1113
.561&
.719
.A75
.1&05
.557
.712
./1&'1
.390;
.q~2
.'123
.573
.727
.883
.392
.cf.lE• -fO 1
.q'lP.
.532
.'190
.'15'1
.581
.710
.6 .. 4
."~5
.!i)O
.691
./126
.'1'6
.'1 .. 3
.676
.8\2
.'101
.'130
.66'1
./l01
.389
.519
.655
.792
.379
.511
.6"6
.76'1
.371
.'503
.363
.1&97
.63!
.771
.1&11
.3 eU
.~19
.502
.6:>0
.7 .. 1
.376
.'189
.... 08
.729
.363
... 78
.353
.469
.!CJA
.S~9
.720
.712
.344
... 61
.'182
.705
.'55
, .. "i6
.562
.671
.3'12
.661
.331
."35
.5 .. 3
.J;!:3
.322
... 27
.536
.646
.326
.'117
.51'1
.613
.314
.407
.505
.605
.305
.399
.'197
.598
.297
.392
... 91
.290
.!!6
.~9,"
.~~7
.291
.H6
.282
.36 9
.=4~~
.n5
.362
.4 C,!
2
5
6
12 1.0
1.5
2.0
2.5
1.166
1.292
1.1127
1.568
14 1.0
1.3
2.0
2.5
.807 .735 .676
1.137 1.Cll
.8'19
.77/1
.903
.833
1.2"6 1.105 .9°2
.9 .. 4
.892
1.~~2 1.212 1.0'17 1.010
1.~~q, 1.327 1.213 1.128 1.06" 1.015
7
.629
.73'1
.851
.977
8
.~4?1
.... 9'1
./119
.""6
.5'19
.670
.702
.921
.~ .. 6
.~Oll
.770
.qOD
.736
.B69
.63C?
,717
.5~1
.71}f,
.P"'~
• 357
."'H
.~2e
.7&7
16 1.0
1.5
2.0
2.5
.759 .715
1.115 .992
.81'1
1,210 1.013 .960
.670
.600
1.31~ 1.165 1.051
.963
.895
1,"19 1.265 1.150 1.065 1.000
.656
.74'1
.842
.949
.61)9
.699
.801
.910
.570
.663
.767
.A7B
.537
.633
.739
.85 2
.510
.6nll
.717
.A31
.467
.569
.661
.HIl
."35
.541
.655
,774
18 1.0
1.5
2,0
2.5
.979
,866
.774
1.0'111
./1 .. 6
.9"16
1.11l1 1.C"1I
.926
1.271 1.12q 1.C14
1.367 1.21S 1.lnl 1.015
.700
.774
./157
.949
.6 .. 1
.71'1
.804
.1197
.593
.672
.761
.857
.'153
.1',35
.726
.1124
.521
.604
.6'}8
.7?8
.493
.579
.1',74
.776
.4'19
.638
.7'12
."1"
.510
.611
.717
.3ql
... 117
.'191
.,;98
.371
,4'10
.'575
.6A3
20 1.0
1.5
2.e
2.5
1.0/1'1 .967 .6'15
1.\51l 1.02/\
.917
1,23':' 1.0'1/1
.9P6
1.32'1 1.176 1.01'.2
.H\O\
.629
.636
.772
.656
.560
.650
.729
.81'1
.5'10
.612
.5n7
.'179
.754
.827
.908
.5~1
.~55
.781
.61'.14
.75'1
.6'10
.731
.401
."8'1
.575
.670
.376
."61
.55'1
.651
.3!'>5
..... 3
.5'8
,636
.339
.429
.1'.93
.'135
.5114
.603
.696
22 1.0
1.5
2.0
.570
.632
.702
.779
.530
.0;94
.'.66
.7'15
.496
.51;2
.6:'17
.717
.'168
.536
.612
.69'1
.423
."94
.57'1
.659
.389
."63
.300
.3~4
.l,d~2
.573
.413
.565
.1~66
.~12
.326
.406
.'193
.564
.3q~
.632
.3 .. 2
,421
.507
.596
.312
.:'45
.363
."'10
.5:14
2.~
1.073 .957
1.13 0 1.012
1.21.~ 1.C7/:
1.283 1.144
.5~9
.s~;:.
.~46
24 1.0
1.5
2.0
2.5
1.064 .950
1.12'1 .999
1.190 1.0'55
1.2~0 1.117
."59
.520
.589
.663
."13
."77
.549
.1'.25
.379
..... 6
.520
.599
.3 .. 3
."22
... 98
.3"11
,'In3
,'1'1
.389
,t461
.~~9
.210
.3 .. 9
.'132
.~79
.5"'3
,550
.289
.365
.'1 .. 7
.'531
.279
.356
."~1
.300
.37'5
.'156
.539
.'523
.~17
.263
.!t+3
."26
."i12
26 1.0
1.5
2.0
2.5
.943
.969
1.~71 1.038
1.235 1.09'1
.451
.506
.40':
.3?2
.5hl)
."07
."77
.'\'\1
.3~7
.5~"
.371
.'131
.'199
.572
.~L.;Lt
.5":~
.30'5
.H2
."'10;
.'\21
.2 9 1
.359
.U33
.219
.3'19
.42'1
.501
.26<;
.3 4 0
,"16
.49'1
.26 C·
.332
.409
.315
.37'1
.440
.510
.:197
• ~~8
.2B2
.42~
• 'I
,260
.325
.396
•• 96
,4e5
.270
.33'1
.40'1
,476
,3011
.3"3
.42'1
.'90
.347
,'109
... 74
.275
.263
.322
• ~·e7
.4'3'1
1.0%
1.110
.762
.P26
./\97
.975
.845 '.752
.&09
.9~1
.873
.?F.2
.343
1.0~0
.678
.737
.875
.618
.679
.747
.621
.670
.723
.782
.8"7
.610
.665
.726
.793
.561
.617
.661
.750
.'121
.!'>7"
.'1,.7
.~Ll4
.614
1.0~'I
.7 .. 4
.796
.1\53
.916
,71'1
.6~7
.8"11
.877
.'}27
.9P2
.738
.755
.1137
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.663
.711
,603
.652
.701\
.76'1
.554
.605
.662
.725
.513
.51'.6
.1',25
.F.9n
.479
.61'.1
.637
.44'.
.495
.t.C'."
.473
.522
.578
.837
.81l6
.943
.(\Q2
• '7(>S
.1'2'1
28 1.0
1.5
2.0
2.5
1.0"9
1.099
.937
.97'1
1.025
1.213 1.075
.825
.867
.'H3
.964
.732
.775
• 657
.701
.~22
.7~O
.87"
30 1.0
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2.0
2.5
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.820
1.08~
.B~9
.727
.767
.1\10
,0';8
1.1~'1
.933
.971
1.1 .. 0 1.013
1.195 1.059
.902
.9"13
.~'.. 7
.548
.'59'1
.E'I7
.70"
.'\07
.'i55
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.en
• 7'.. 9
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.731
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.:,52
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... b3
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.... 35
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."01
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.4sa
.""9:
.5'4
.~.~d
.~25
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.4U4
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.~45
~
..
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.~~l.l
.41&9
.'176
.699
.~71
.31~
.42~
.30!!
."1'1
.52'1
.6'1e
.1,3~
.~94
.~24
.1f'\F:
.253
.~26
.4ij3
."+eA.
.4-~2
• .,6S
.251
.31P
• "0
_r. •
."e2
.2 ....
.311
.253
,313
.378
.24'"
.305
.311
.236
,'I4~
.~4("l
•
.!.,Q!
.U':E.29~
.~65
~
34
'-J
'-J
78
value k
o
and a(k ) = Pr(fU
0
l
+ U > k ) was evaluated.
z-
0
Iteration pro-
ceeded, producing successive values kl,.••• ,k , until Ik - k _ ' <
n
n
n l
.00005.
Then k
n
value k-.
n
was truncated to three decimals--call the truncated
The values a(k-) and a(k+)
n
n
= a(k-n
+ 0.001) were evaluated.
If a(k-) was closer than a(k+) to the desired a-level, k- was chosen
n
n
n
as the tabular value, k; otherwise k+ was chosen.
n
The rounding of k to
three decimals produces an error of la(k)-al, which does not exceed
0.00020 if a
a
=
=
.01 is the desired level of the test and 0.00060 if
.05 is the desired level.
For most values the errors are substan-
tially smaller, usually less than one-half the stated maximum for the
significance level.
For values not included in the tables, interpolation is sometimes useful.
Interpolation over values of b does not lead to good
approximations, but interpolations over the parameters aI' a , and f
2
does result in quite accurate values for k.
larger values of f is also feasible.
of a
l
and a
2
Extrapolation for some
Extrapolation for larger values
was not done since it seems clear from the values in
Table 5.1.1 that extrapolation to values of a
l
and a
2
which are reason-
ably close to the maximum tabular value of 30 should yield good approximations to the true value of k.
Table 5.1.2 presents values of
parameters for which interpolation or extrapolation was performed.
(Since the
parameters a
l
and a
2
are interchangeable if f
results for certain examples of interpolation over a
sponding examples of interpolation over a .)
2
= 1,
the
apply to corre-
l
The asterisk indicates
the parameter on which interpolation or extrapolation was performed.
The value of k, say k, which was found by interpolation or extrapolation
"
is also given along with the corresponding error, la(k)-al.
79
TABLE. 5 .1. 2
ESTIMATED CRITICAL VALUES FOR TYPE D AND TYPE F TESTS
FOR CERTAIN INTERPOLATED AND EXTRAPOLATED VALUES
OF THE PARAMETERS a , a , ANDf
2
1
A
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.05
0.05
0.05
0.01
0.01
0.01
0.01
0.05
0.05
0.05
0.01
0.05
0.05
0.05
0.01
0.01
0.01
b
a
1.0
2.0
1.0
2.0
1.0
1.0
2.0
1.0
2.0
1.0
2.0
1.0
2.0
1.0
2.0
1.0
1.0
1.0
1.0
2.0
2.0
2.0
2.0
1.0
1.0
2.0
1.0
1.0
1.0
1.5*
1.5*
1. 5*
1.5*
1.5*
29.0*
29.0*
29.0*
29.0*
29.0*
29.0*
29.0*
30.0
30.0
30.0
1.0
1.0
30.0
30.0
30.0
1.0
1.0
30.0
30.0
1.0
1.0
1.0
30.0
30.0
*The
1
a
2
1.0
1.0
1.0
30.0
30.0
1.0
1.0
1.0
1.0
30.0
30.0
30.0
1. 5*
29.0*
29.0*
1.0
30.0
1.0
30.0
30.0
1.0
30.0
30.0
30.0
1.0
1.0
30.0
1.0
30.0
f
1.00
1.00
2.50
1.00
2.50
1.00
1.00
2.50
2.50
1.00
1.00
2.50
2.50
2.50
2.50
1.25*
1. 25*
1.25*
1.25*
1.25*
2.25*
2.25*
2.25*
2.25*
3.00*
3.00*
3.00*
3.00*
3.00*
k
1. 770
1.875
3.102
1.063
2.414
1.040
1.109
1.171
1.349
0.209
0.304
0.407
1.124
0.279
0.436
2.092
1.280
1.055
0.234
0.266
2.997
2.256
0.518
0.255
3.450
3.708
3.001
1.209
0.464
A
ICl(k)-al
0.00015
0.00008
0.00011
0.00027
0.00054
0.00020
0.00009
0.00005
0.00012
0.00006
0.00003
0.00010
0.00005
0.00041
0.00036
0.00001
0.00004
0.00001
0.00016
0.00042
0.00010
0.00073
0.00000
0.00025
0.00042
0.00040
0.00042
0.00032
0.00024
asterisk signifies the parameter for which the interpolation or extrapolation was performed.
80
Interpolation was done using Newton's forward-difference and backwarddifference formulas (Conte [1965]).
difference formulas.
Extrapolation was also done using
For interpolation on a
l
and a , five known values
2
of k were used as interpolating points; for interpolation and extrapolation on f, all four known values of k for specified aI' a , b and a
2
were used.
Programming for the critical values in Table 5.1.1 was done in
PL-l and the PL-l Scientific Subroutine Package program BDTR was used
in the computations.
BDTR returns the incomplete beta function ratio
I (b,a) for given x, b, and a.
x
As installed on the University of North
Carolina Computation Center's IBM 360/75, BDTR returns incorrect values
if b
= 1,
a
~
1, and x
~
1.
It was possible to avoid using this sub-
routine in any such case by substituting the equivalent value of
11
-x
(a,b) for I (b,a) as necessary.
x
5.2. Numerical Illustration
In this section the use of the type D and type F tests in
analysis will be demonstrated in an application to data supplied by a
pharmaceutical company.
Data on the effects of six parallel treatments
on diastolic blood pressure measured in sitting position in two hypertensive studies are used.
The cell frequencies of the data are as
follows:
TABLE 5.2.1
DISTRIBUTION OF PATIENTS AMONG SIX HYPERTENSION
TREATMENTS IN TWO CLINICS
Treatment
4
1
2
5
6
3
1
10
9
7
9
8
9
N
i
52
2
11
11
9
9
10
12
62
Clinic
81
The observation x
value (Y
ijk
was obtained as the difference of the pre-treatment
ijk
) and the average of post-treatment values collected during
four visits (z, 'k).
Therefore, the average decrease in diastolic blood
~J
pressure of the jth patient at clinic i receiving treatment k is given
by
(i
1,2; j
1, ••• ,N.; k = 1, ••• ,6).
~
Consider the model
X'ijk
~ik
where lli is the overall mean at clinic i,
effect at clinic i (after
Let
~i
~eparameterization)
is the k
and
E,
th
treatment
'k is the error.
~J
be a N x 1 vector of the observations x
for clinic i and
ijk
i
=
let fi
(5.2.1)
= lli + ~ik + £ijk'
(lli'~i1' ••• '~i5)'.
Then (5.2.1) can be rewritten as
Xi
== A.~.
E.
_
_J._1 + -1
th
th
where the m row of A. corresponds to the m row (patient) of X.
~~
~~
(clinic i) and is
1
1
0
0
0
0
1
-1
1
1
1
1
0
1
0
0
0
-1
0
0
1
0
0
-1
0
0
0
1
0
-1
0
0
0
0
1
-1
if
if
if
if
if
if
patient
patient
patient
patient
patient
patient
m
m
m
m
m
received
received
received
received
received
to. received
treatment
treatment
treatment
treatment
treatment
treatment
1
2
3
4
5
6
and ~i is the vector of errors for clinic i.
In each clinic the hypothesis tested is
H
o
where
~
=
[25x1:
C
~ M
""'" _1
_ == Q
__ versus HI:
!5x 5] and
~ ==
1.
~ -0
-C~.
..... 1 M
-
Assuming that each error vector
has an Ni-variate normal distribution with mean zero and Ni x Ni
82
variance-covariance matrix a 2 I, under"H
F.
1
=
-1
(N i - 6) H.E.
/5, i
-1-1
= 1,2,
the statistic
o
has the central F distribution with 5
and (N i - 6) degrees of freedom, where
~i
Section 1.1, are scalars in this problem.
and
~i'
defined as in
In this example
F (5,46)
l
= 2.3964,
PI
= 0.05168;
F (5,56)
= 2.7523,
P2
= 0.02716;
and
2
the numbers in parentheses indicate the degrees of freedom.
The
statistics were calculated using the MGLM program, installed at the
Triangle Universities Computation Center, and the P-values were calculated by the TSO program CFrDIS, also at TUCC.
Since M = 1, the null hypothesis to be tested is, for each
clinic, H
a
.
A. = CC =
- -1
-1
0, i
= 1,2. For this problem the noncentrality
parameters are
8
i
C,]-l C s/a~
s.)'[C(A~A.)-l
= (C_ -1
- ...1-1
].
-
so that
, )-1, -1 2
V. _
- [ C(A.A.
C] /a .
-1
... -1-1
...
(5.2.2)
i
For each clinic the maximum likelihood estimate of
was substituted for a~.
a~,
1
It is not necessary to assume that o~ = o~,
as it would be if a single model were fitted to the data.
For the
2
0
= 0 2 versus
HI: 0 22
I 02
Problem in this example, the test of H:
0
1
21
leads to rejection of H at a = .10.
o
In order to calculate the statistic flU
l
+ f 2U2 for the type F
test it is necessary to find the values of U , U , f , and f 2 •
l
2
l
Each Ui
83
is a simple transformation of the F-distributed statistics F..
For
1
this example, since b = 5/2 = 2.5, a
1
=
46/2 = 23.0, and a
2
=
56/2 =
28.0,
bF/a 1
2.5(2.3964)/23
U1 = 1+bF /a = 1 + 2.5(2.3964)/23
1 1
0.2067
and
bF/a 2
2.5(2.7523)/28
= 0.1973.
U =
=
1 + 2.5(2.7523)/28
2
1+bF/a 2
In order to find the f.' s it is necessary to calculate y. = tr(V.),
_1
1
1
i
= 1,2,
where the V.
's are of the form given in (5.2.2).
_1
the matrices [~(~l~i)
-1
~']
-1
= 1,2,
, i
are provided by MGLM can be used.
[~(~I~i)
,
(A.A )
-1- i
-1
-1
•
~']
To determine
-1
the 6x 6 matrices (~~~i)
which
Since ~
=
[~5x1:
!5x5]'
can be found by deleting the first row and column of
The resulting 5x5 matrices were inverted to find
18.981
9.000
18.000
9.038
9.000
15.922
9.000
9.000
9.000
18.000
9.019
9.000
8.961
9.000
16.981
22.983
11.983
22.983
11.951
11. 951
20.854
11. 951
11. 951
11.854
20.854
11. 967
11.967
11.902
11.902
21. 935
[C(A'A )-l C,]-l =
- _1_1
-
and
[C(A' A ) -l C' ]-1 =
- _2_2
-
"2
"
"
MGLM also provides NiO i , so that O~ and 0 22 can be
"
N1 O~
"
=
of =
2405.379 ==> "
N2 o~ == 2057.861
ea~i1y
2405.379/52 = 46.257;
"
> O~ = 2057.861/62
33.191.
found:
84
It follows, therefore, that
= 87.884/46.Z57
= 1.900
and, similarly,
YZ = tr(~Z) ~ 3.30Z.
The parameters Y and YZ having been found, it is possible to calculate
l
=
(Z3.0 + Z.5)(1.900)/Z.5
= 19.380;
f Z = (a Z + b) YZ/b = (Z8.0 + Z.5)(3.30Z)/2.5
40.284.
~l
(al + b) Yl/b
=
Finally it is possible to calculate the statistic for the type F test.
Since f Z > f l , the form fZUZ/fl + Ul = k is used.
It follows from the
above that
k
= 40.Z84
=
Since k
~
(0.1973)/19.380 + 0.Z067
Z.079 (0.1973) + 0.Z067 = 0.617.
1.0 and kfl/f Z
~
1.0, a program using the algorithm to provide
significance levels for Case 3 situations can be used to determine that,
for this problem,
Pr(Z.079 U + U
z l
~
0.617)
=
.00869.
Therefore, the overall test is significant at a
=
.01 whereas neither
of the individual F statistics, F and FZ' is significant at that level.
l
By contrast, Fisher's statistic for this problem is
-2 log (PlP Z)
= -21og
(0.05168 x 0.OZ716) = 13.1374.
This statistic is
distributed as a chi-squared with four degrees of freedom under H.
o
By
85
use of CFTDIS it can be found that
Pr~x: ~ 13.1374)
=
.01062.
For this problem, the type F test is found to be more sensitive than
Fisher's test to deviations from the null hypothesis, even though there
are only two statistics combined.
This result is of particular interest
in light of the results of Section 3.3, where it was shown that the
type D and type F tests are more efficient assuming that there are a
large number of statistics to be combined.
The above application illustrates the method of calculating a
statistic for the type F test.
Since the type D test coincides with
the type F test if indeed the V. are proportional, the above method of
-~
calculation, with y.
~
=
tr(V.),can be used for type D tests also.
-~
CHAPTER VI
SUMMARY AND SOME SUGGESTIONS FOR FURTHER RESEARCH
In this dissertation two locally most powerful procedures were
found for combining independent statistics which arise in the analysis
of certain multiparametric and multivariate problems.
The type D test
for combining beta-distributed statistics was found for the case of a
balanced design; the type F test was introduced for use in the case of
an unbalanced design.
The overall statistic for type D and type F
tests is, in each case, a linear combination of the beta-distributed
statistics to be combined.
The exact distribution of the statistic
was found in the two-sample case; the accuracy of using normal approximations for the general s-sample case was also considered.
It was
shown that for local alternatives, the type D test is more efficient
than Fisher's test for combining statistics, according to Bahadur
asymptotic relative efficiency, if there are a large number of smallsample statistics to be combined.
Under the same conditions, it was
shown that the type F test is more efficient than Fisher's test
according to
ciency.
a~
average measure of Bahadur asymptotic relative effi-
In a numerical example, the theor~tical developments were
applied to some phannaceutical data.
Several areas of future research may be suggested.
It would
be useful to work out the details of the exact distribution
of the type D and type F statistics for larger values of s.
As noted
87
earlier, however, this task would be tedious and the calculations
necessary to find critical values would. be expensive.
It would also be useful to develop the details of the distributions of the type D and type F statistic if the statistics to be combined have chi-squared distributions.
If s = 2, the exact distribution
should be reasonably simple to calculate, with more complicated results
for larger values of s.
Another area of further research is consideration of the
extensions mentioned in Section 3.4.
The exact distributions of the
combined test statistics mentioned there may be sufficiently complicated
to warrant study of approximations.
As mentioned earlier, Littell and Folks have shown that if
there are a finite number of large-sample statistics to be combined,
Fisher's method is more efficient than other methods according to
Bahadur asymptotic relative efficiency; the locally most powerful combination procedures were shown to be more efficient if there are a
large number of finite-sample statistics to be combined, for local
alternatives.
Research into the relative efficiencies of Fisher's
test and the locally most powerful tests for combining an intermediate
number of medium-sized sample statistics for local and non-local
alternatives would be illuminating.
APPENDIX A
USEFUL MATHEMATICAL RESULTS
Lemma A.l.
If U. has a beta distribution with density
1
e.
Su.1
specified
1
in (1.1.2), then (2.3.4) and (2.3.5) are equivalent.
s
Proof.
In (2.3.4)
II
h=l
form of (1.1. 2).
eh
Su
is the product of s infinite series of the
h
Since each of these series is a positive term con-
vergent series, each is absolutely convergent and (Taylor and Mann
[1972, p. 635]) the order of summation can be interchanged.
It remains
to be shown that each of the infinite sums can be taken outside the
integral.
If b
~
1 and a
j
~
1, then the density of each U is uniformly
j
If b = 1/2 then for every j,
bounded and the interchange is immediate.
the density of U. goes to infinity as U.
J
J
the density goes to infinity as U t 1.
j
dU
j
~
O.
J
However, since for all j
= 1, the dominated convergence theorem (Cramer [1945, p. 45])
applied.
Therefore, each summation can be taken outside the
integral and the lemma holds.
Lemma A.2.
If a. = 1/2 for some j,
o -<
are as defined in (1.1.3), then
u.1<- 1, i
89
u
!
eO eO
,w
1
u
u
2
dUldu Z =
r
(1)
u
2
u
=
(2)
2
r
U(2)(u )
1
Z
r
ui 1 ) (u )
eOu eOu
1
Z
(2)
l
u(2)(u)
Z
1
r
U~l)(u1)
(1)
u
l
du du
l 2
Z
eOu 13u0
1
dU du ,
2 l
2
where the limits of integration depend on the boundary w
Proof.
As long as the function
w.
is integrable in w, the result
holds (Taylor and Mann [1972, p.
euo ,
c
The beta densities, 13
0
u
and their product, are integrable when w
c
w.
and
l
2
Lemma A.3.
00·
Let f(x) = Ln=O anx
convergence R > O.
n
be a power series with radius of
The function f is continuous in the open interval
of convergence of the series.
If p and q are points in [O,R), p < q,
then
r
p
q
~oo
f(x)dx = un=O
an
n+l
0+1
.;Iq'--_ _
-~P-_
n+l
.•
If q = R, the result holds if the right hand side is convergent.
Reference.
Discussion.
Taylor and Mann [1972, p. 665].
In all the instances in which this lemma is used in the
dissertation, the series is a Taylor series in which
a
n
= fen) (O)/n!,
where fen) denotes the nth derivative of f.
90
Lemma A.4.
If 0
~
~
x
c/d and if n > 0, then
where
'*
n
{n, if n is an integer;
~,otherwise.
=
Reference.
Taylor and Mann [1972, p. 632].
Lemma A.5.
Let 0
1, b > 0, and a > O. If
S (b,a) = f x u b-1 (l-u) a-I du
~
~
x
x
0
is the incomplete beta function and
I (b,a)
x
Sx (b,a)/S(b,a)
=
is the incomplete beta function ratio, then
(a-I) '*
Sx (b,a)
= L
S(b,a)
=L
i=O
1
=
(a-I) '* a-I
i=O
1.
~
x b-1
x
0
(l-u)
i=O
a-I
(A.5.1)
,and
du
(A. 5.2)
1.
(a-I) '* a-I (_l)i b+i
Q(b,a) L
( i ) b+i
x
,
By Lemma A.3 and A.4,
S (b,a) = f u
(_l)i
( " ) b+"
where (a-I) '* is defined in Lemma A.4.
Proof.
b+1."
x
1.
1
I (b,a)
x
(-1) i
(a~ ) b+1."
(A.5.3)
91
(a-I) *
=L
i=O
i
( a-I) H2.. x b+i
i
b+i
This verifies the first claim.
the second claim follows by setting x
=
S(b,a)., so
= 1 in the first result.
third claim follows by definition of I (b,a).
x
Lemma A.6.
f\ (b,a)
By definition,
The
The completes the proof.
If p, q, a, and b are real numbers such that q > p, b > 0,
and a > 0, then
f
q
(u_p)b-l (q_u)a-l du
(q-p)
a+b-l
S(b,a).
p
Discussion.
and q
= 1,
Although the standard beta distribution specifies p
=0
a family of beta distributions is given by density functions
of the form
(Johnson and Kotz, Vol. 2 [1970, p. 37]).
a standard
Lemma A.7.
If V
=
(U-p)/(q-p), V has
beta distribution.
If x
~
1, b > 0, a > 0, and c > 1, then
CA.7.1)
converges.
Proof.
vergent.
If a is an integer the series is finite and, therefore, cona-I
If a is not an integer, then notice that ( i )(-1)
positive or negative for all i > (a-I).
i
is either
If [al is even, the terms are
positive; if [a] is odd, the terms are negative, where [a] is the
greatest integer in a.
If [a] is odd, multiply the .series by (-1),
92
which will not affect the convergence or divergence of the series.
Therefore, without loss of generality, .assume that the series is a
positive term series in the tail.
It will be shown that (A.7.l) converges for non-integral a by
showing that
00
I~
(a-I) (_l)i xb+i Q
(b+i)
~=O i
~l/x'c
I<
where M and M are positive constants.
l
2
00
+
IL
i=[a]
00
M
M I~
(a-I) (-1)
1 + 2 ui=[a] i b+i
i
I
(A 7 2)
••
By Lemma A.5,
.
I
(a-I
i )(-1) i x b+i f3 l / x (b+1,c).
(A. 7 • 3)
Next, by Lemma A.5,
'Ie
(c-l)
(l)j] •
+ L (c-l) (-l)j b+i
b+i+j
x
j=[e]
j
(A. 7 .4)
93
[c-l]
.
E (c-l) (l)J,
j=O
j
x
The first series in (A.7.4) is finite and bounded by M
3
which is finite.
If c is an integer, the second series in (A.7.4) is O.
00
Otherwise, recall that
Mann [1972, p. 632]).
E (c~l)(_l)j converges if c-l > 0 (Taylor and
J
j=O
00
\"'L.
(c-l) (-l)j converges. Since
Therefore,
j= [c] j
all terms in this series are of the same sign,
00
00
j b+i
1·
IE
(c- )(-1) b+i+j (;z)JI < IE
j=[c] j
j=[c]
1
(Cj 1 )(_l)J. I =
4 <
M
00
•
(A.7.5)
Let M = M + M , which is finite.
2
3
4
Then
Let M be the first term on the right hand side of (A.7.3).
l
M <
l
Clearly
Then (A.7.2) follows from (A.7.3) in light of (A.7.6). Since
00
a-I (_l)i
,
~, M and Ei=[a]( i )b+i
are
all
finite,
the lemma holds as long as
2
c > 1. (Note that in all instances when the lemma is used in the text,
00.
c > 1.)
Lemma A.8.
If a > 0, b > 0, c > 0 and x
~
1,
(a-I) * a-I (_l)i b+i
i:O ( i )b+i
x
f3 l / x (b+i+1, c)
converges.
Proof.
The proof proceeds on the principles of the proof of the
previous lemma.
However, (A.7.4) is replaced by
94
x
b+i
Sl/x(b+i+l,c) <
IiC-l] (c-l)
(-l)j
J. b+i+l+J·
j=O
(!.)j+ll
x
(c-l) * c-l
(-l)j
+ Ir
( j )b+i+l+j
j=[c]
I·
(A.8.l)
If c is not an integer, the second term on the right-hand side is the
tail of the series
IX)
c-l (-l)j
r (J. )b+i+l+j = S(b+i+l,c),. which is a decreasing
j=O
function of i and, therefore, is bounded.
(A.8.l) is set equal to M <
2
i
IX)
Ir
i=O
00,
If the right-hand side of
it is then clear that
b+i
xb+~ Sl/x(b+i+l,c) I <
(a-il) (-1)
~
i
00
M
1
+
M
Ir
2 i=[a]
(a-:-l) (-:)
~
b+~
I,
(A. 8.2)
where M is the absolute value of the first a terms of the series on
l
the left-hand side.
Since the series on the right-hand side is the
tail of the convergent series (A.S.2), (A.8.2) is finite and the lemma
holds.
Lemma A.9.
If I (b,a) is an incomplete beta function ratio defined in
x
Lemma A.S, then
I x (b,a) = 1 - 11-x (a,b'.
{
Reference.
K. Pearson [1968, p. vii].
BIBLIOGRAPHY
Anderson, T. W.
Analysis.
[1958].
An Introduction to Multivariate Statistical
Wiley, New York.
Bahadur, R. R.
[1960]. Stochastic comparison of tests.
Statist. 31, 276-295.
Ann. Math.
Bahadur, R. R. [1967]. Rates of convergence of estimates and test
statistics. Ann. Math Statist. 38, 303-324.
Bhattacharya, N. [1961]. Sampling experiments on the combination of
independent x2-tests. Sankhya A 23, 191-196.
Birnbaum, A.
[1954].
Combining independent tests of significance.
J. Amer. Statist. Ass. 49, 559-574.
Birnbaum, A. [1955]. Characteristizations of complete classes of
tests of some multiparametric hypotheses, with applications
to likelihood ratio tests. Ann. Math. Statist. 26, 21-36.
Conte, S. D. [1965].
New York.
Elementary Numerical Analysis.
McGraw-Hill,
Cramer, H. [1946]. Mathematical MethodS of Statistics.
University Press, Princeton.
Princeton
Davies, R. B. The weighted combination of independent normal test
statistics. (Unpublished).
Fisher, R. A. [1932]. Statistical Methods for Research Workers
(4th and Later Editions). Oliver and Boyd, London.
Fisz, M.
[1963].
Probability Theory and Mathematical Statistics.
Wiley, New York.
Good, I. J. [1955]. On the weighted combination of significance
tests. J. R. Statist. Soc. B 17, 264-265.
[1969]. Introduction to Matrices with Applications
in Statistics. Wadsworth, Belmont, California.
Graybill, F. A.
Gupta, A. K. [1971]. Noncentral distribution of Wilks' statistic in
MANOVA. Ann. Math. Statist. 42, 1254-1261.
96
Hill, A. B. [1971]. Principles of Medical Statistics (9th Edn.).
Oxford University Press, New York.
Isaacson, S. L. [1951]. On the theory of unbiased tests of simple
statistical hypotheses specifying two or more parameters.
Ann. Math. Statist. 22~ 217-234.
Johnson, N. L. and Kotz, S. [1970]. Continuous Univariate Distributions~ Vols. 1 and 2.
Houghton Mifflin, Boston.
Kincaid, W. M. [1962]. The combination of test statistics based on
discrete distributions. J. Amer. Statist. Ass. 57~ 10-19.
Lancaster, H. O. [1949]. The combination of probabilities arlslng
from data in discrete distributions. Biometrika 36~ 370-382.
Lancaster, H. O. [1961]. The combination of probabilities: an
application of orthonormal functions. Austral. J. Statist.
20-33.
Liptak, T.
[1958].
On the combination of independent tests.
3~ 171-197.
3~
Magyar
Tud. Akad. Mat. KutatO Int. Kozl.
Littell, R. C. and Folks, J. L. [1971]. Asymptotic optimality of
Fisher's method of combining independent tests. J. Amer.
Statist. Ass. 66~ 802-806.
Littell, R. C. and Folks, J. L. [1973]. Asymptotic optimality of
Fisher's method of combining independent tests II. J. Amer.
Statist. Ass. 68~ 193-194.
Mantel, N. [1963]. Chi-squared tests with one degree of freedom;
extensions of the Mantel-Haenszel procedures. J. Amer. Statist.
Ass. 58, 690-700.
Morrison, D. F. [1967].
Hill, New York.
Multivariate Statistical
Methods~
McGraw-
Naik, U. D. [1969]. The equal probability test and its applications
to some simultaneous inference problems. J. Amer. Statist.
Ass. 64~ 986-998.
Neyman, J. and Pearson. E. S. [1936]. Contributions to the theory
of testing statistical hypotheses, part I. In Statistical
Research Memoirs~ Vol. 1. Issued by the Department of Statistics, University of London, University College, pp. 1-37.
Neyman, J. and Pearson, E. S. [1938]. Contributions to the theory of
testing statistical hypotheses, parts II and III. In Statistical Research Memoirs~ Vol. 2. Issued by the Department of
Statistics, University of London, University College, pp. 25-57.
97
Oosterhoff, J. [1969]. Combination of One-Sided Statistical Tests.
Tract No. 28, Amsterdam Mathematical Centre, Amsterdam.
Pearson, E. S. [1938]. The probability integral transformation for
testing goodness of fit and combining independent tests of
significance. Biometrika 30~ 134-148.
Pearson, E. S. [1950]. On questions raised by the combination of
tests based on discontinuous distributions. Biometrika 37~
383-398.
Pearson, K. [1933]. On a method of determining whether a sample of
given size n supposed to have been drawn from a parent population having a known probability integral has probably been
drawn at random. Biometrika 25~ 379-410.
Pearson, K. (Ed.) [1968]. Tables of the Incomplete Beta Function
(2nd Edn.). Cambridge University Press, London. (1st Edn.,
1934).
Pi11ai, K. C. S. and Young, D. L. [1971]. On the exact distribution
of Hote11ing's generalized T~. J. Multivariate Anal. 1~ 90107.
[1971]. Analysis
and Design of Certain Quantitative Multiresponse Experiments.
Roy, S. N., Gnanadesikan, R. and Srivastava, J. N.
Pergamon Press, New York.
Schaafsma, W. [1968]. A comparison of the most stringent and the
most stringent somewhere most powerful test for certain problems with restrictive alternative. Ann. Math. Statist. 39,
531-546.
Sen, P. K. [1968]. Robustness of some nonparametric procedures in
linear models. Ann. Math. Statist. 39~ 1913-1922.
Siotani, M. [1971]. An asymptotic expansion of the non-null distribution of Hote11ing's generalized T~-statistic. Ann. Math.
Statist. 42~ 560-571.
Sugiura, N. and Fujikoshi, Y. [1969]. Asymptotic expansions of the
non-null distributions of the likelihood ratio criteria for
multivariate linear hypothesis and independence. Ann. Math.
Statist. 40, 942-952.
Taylor, A. and Mann, W. R. [1972]. Advanced Calculus.
Publishing, Lexington, Massachusetts.
Tippett, L. H. C. [1931].
Norgate, London.
The Methods of Statistics.
Xerox College
Williams and
98
Van E1teren, Ph. [1958]. On the combination of independent two sample
tests of Wilcoxon. BuZZ. Intern. Statist. Inst. 37, 351-361.
Van Zwet, W. R. and Oosterhoff, J. [1967]. On the combination of
independent test statistics. Ann. Math. Statist. 38, 659-680.
Wa1d, A.
[1943]. Tests of statistical hypotheses concerning several
parameters when the number of observations is large. Trans.
Amer. Math. Soc. 54, 426-482.
Wallis, W. A. [1942]. Compounding probabilities from independent
significance tests. Econometrica 10, 229-249.
Wey1, H.
[1939].
461-472.
On the volume of tubes.
Amer. J. of Math. 61,
Wi1k, M. B. and Shapiro, S. S. [1968]. The joint assessment of
normality of several independent samples. Technometrias 10,
825-839.
Wilkinson, B. [1951]. A statistical consideration in psychological
research. Psych. BuZZ. 48, 156-158.
Zelen, M. and Joel, L. S. [1959]. The weighted compounding of
independent significance tests. Ann. Math. Statist. 30,
885-895.
two